View source for Biography:Jamshīd al-Kāshī

{{short description|Persian astronomer and mathematician}}
{{Infobox religious biography
| religion = [[Religion:Islam|Islam]]
| occupation = Persian [[Religion:Muslim|Muslim]] [[Philosophy:Scholar|scholar]]
| era = Islamic Golden Age-Timurid Renaissance
| image = Sharhe alate rasad 3.jpg
| caption = 
| name = Ghiyāth al-Dīn Jamshīd Kāshānī
| title = al-Kashi
| birth_date = {{circa|1380}}
| birth_place = Kashan, [[Religion:Iran|Iran]]
| death_date = {{death-date and age|22 June 1429|1380}}
| death_place = [[Place:Samarkand|Samarkand]], Transoxania
| ethnicity = Persian
| region = [[Religion:Iran|Iran]]
| Maddhab = 
| school_tradition = 
| main_interests = Astronomy, Mathematics
| notable_ideas = Pi decimal determination to the 16th place<br>Law of cosines
| works = Sullam al-Sama
| influences =
| influenced =
}}

'''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|غیاث الدین جمشید کاشانی}} ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 Kashan, [[Religion:Iran|Iran]] – 22 June 1429 [[Place:Samarkand|Samarkand]], Transoxania) was a Persian [[Religion:Astronomy in the medieval Islamic world|astronomer]] and mathematician during the reign of Tamerlane.<ref>{{cite book |last1=Bosworth |first1=C.E. |title=The Encyclopaedia of Islam, Volume IV |date=1990 |publisher=Brill |location=Leiden [u.a.] |isbn=9004057455 |page=702 |edition=2. impression. |quote=AL-KASHl Or AL-KASHANI, GHIYATH AL-DIN DjAMSHlD B. MASCUD B. MAHMUD, Persian mathematician and astronomer who wrote in his mother tongue and in Arabic.}}</ref><ref>{{cite book |last=Selin |first=Helaine |title=Encyclopaedia of the history of science, technology, and medicine in non-western cultures |publisher=Springer |location=Berlin New York |year=2008 |isbn=9781402049606 |page=132 |quote=Al-Kāshī, or al-Kāshānī (Ghiyāth al-Dīn Jamshīd ibn Mas˓ūd al-Kāshī (al-Kāshānī)), was a Persian mathematician and astronomer.}}</ref>

Much of al-Kāshī's work was not brought to Europe, and still, even the extant work, remains unpublished in any form.<ref>[http://www.iranicaonline.org/articles/kasi] ''iranicaonline.org''</ref>

==Biography==
[[File:Selm Alsamaa 3.jpg|200px|thumb|right]]
[[File:Resaleye mohitiye end.jpg|200px|thumb|left|Theorem of Al-Kashi]]

Al-Kashi was one of the best mathematicians in the history of Iran. He was born in 1380, in Kashan, in central Iran. This region was controlled by Tamerlane, better known as Timur.

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 Turkish 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.

Eight years after he came into power in 1409, their son, [[Biography:Ulugh Beg|Ulugh Beg]], founded an institute in [[Place:Samarkand|Samarkand]] which soon became a prominent university. Students from all over the [[Place:Middle East|Middle East]], and beyond, flocked to this academy in the capital city of Ulugh Beg's empire. Consequently, Ulugh Beg gathered many great mathematicians and scientists of the [[Place:Middle East|Middle East]]. 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.

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, probably in 1429. Some scholars believe that Ulugh Beg may have ordered his murder, because he went against Islamic theologians.

==Astronomy==

===''Khaqani Zij''===
Al-Kashi produced a ''[[Astronomy:Zij|Zij]]'' entitled the ''Khaqani Zij'', which was based on [[Biography:Nasir al-Din al-Tusi|Nasir al-Din al-Tusi]]'s earlier ''[[Astronomy:Zij-i Ilkhani|Zij-i Ilkhani]]''. In his ''Khaqani Zij'', al-Kashi thanks the Timurid sultan and mathematician-astronomer [[Biography:Ulugh Beg|Ulugh Beg]], who invited al-Kashi to work at his [[Astronomy:Observatory|observatory]] (see Islamic astronomy) and his [[Organization:University|university]] (see Madrasah) which taught [[Religion:Kalam|theology]]. Al-Kashi produced [[Trigonometric functions|sine]] tables to four [[Sexagesimal|sexagesimal]] digits (equivalent to eight [[Decimal|decimal]] places) of accuracy for each degree and includes differences for each minute. He also produced tables dealing with transformations between [[Coordinate system|coordinate system]]s on the [[Celestial sphere|celestial sphere]], such as the transformation from the [[Astronomy:Ecliptic coordinate system|ecliptic coordinate system]] to the [[Astronomy:Equatorial coordinate system|equatorial coordinate system]].<ref name=MacTutor/>

===''Astronomical Treatise on the size and distance of heavenly bodies''===
He wrote the book Sullam al-Sama on the resolution of difficulties met by predecessors in the determination of distances and sizes of [[Astronomy:Astronomical object|heavenly bodies]] such as the [[Astronomy:Earth|Earth]], the [[Astronomy:Moon|Moon]], the [[Astronomy:Sun|Sun]] and the Stars.

===''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 [[Astronomy:Triquetrum|triquetrum]] and [[Astronomy:Armillary sphere|armillary sphere]], the [[Astronomy:Equinox|equinoctial]] armillary and [[Astronomy:Solstice|solsticial]] armillary of Mo'ayyeduddin Urdi, the [[Sine|sine]] and [[Versine|versine]] instrument of Urdi, the [[Astronomy:Sextant (astronomical)|sextant]] of al-Khujandi, the Fakhri sextant at the Samarqand observatory, a double quadrant [[Azimuth]]-[[Altitude|altitude]] instrument he invented, and a small armillary sphere incorporating an alhidade which he invented.<ref>{{Harv|Kennedy|1951|pp=104–107}}</ref>

====Plate of Conjunctions====
Al-Kashi invented the Plate of Conjunctions, an [[Analog computer|analog computing]] instrument used to determine the time of day at which planetary conjunctions will occur,<ref>{{Harv|Kennedy|1947|p=56}}</ref> and for performing [[Linear interpolation|linear interpolation]].<ref name=Kennedy/>

====Planetary computer====
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 [[Earth:Longitude|longitude]] of the [[Astronomy:Sun|Sun]] and [[Astronomy:Moon|Moon]],<ref name=Kennedy>{{Harv|Kennedy|1950}}</ref> and the [[Astronomy:Planet|planet]]s in terms of elliptical orbits;<ref>{{Harv|Kennedy|1952}}</ref> the [[Earth:Latitude|latitude]]s of the Sun, Moon, and planets; and the [[Astronomy:Ecliptic|ecliptic]] of the Sun. The instrument also incorporated an alhidade and [[Ruler|ruler]].<ref>{{Harv|Kennedy|1951}}</ref>

==Mathematics==

===Law of cosines===
In [[Social:French language|French]], the [[Law of cosines|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 for [[Triangulation|triangulation]].<ref>{{Cite book |url=https://books.google.com/books?id=JrslMKTgSZwC&pg=PA106&lpg=PA106&dq=al+kashi+law+of+cosines#v=onepage |title=The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics |last=Pickover |first=Clifford A. |date=2009 |publisher=Sterling Publishing Company, Inc. |isbn=9781402757969 |language=en |page=106}}</ref> His other work is al-''Risāla al''-''muhītīyya'' or "The Treatise on the Circumference".<ref>{{Cite journal|last=Azarian|first=Mohammad K.|date=2019|title=An Overview of Mathematical Contributions of Ghiyath al-Din Jamshid Al-Kashi [Kashani]|url=http://mir.kashanu.ac.ir/article_88765_5a96aa80d8bb6f750c8824481b79806b.pdf|journal=Mathematics Interdisciplinary Research|volume=4|issue=1|pages=|doi=10.22052/mir.2019.167225.1110|via=}}</ref>

===''The Treatise of 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 {{pi}}, which was the most accurate approximation of sin 1° in his time and was not surpassed until Taqi al-Din in the sixteenth century. In [[Algebra|algebra]] and [[Numerical analysis|numerical analysis]], he developed an [[Iterative method|iterative method]] for solving [[Cubic equation|cubic equation]]s, which was not discovered in Europe until centuries later.<ref name=MacTutor/>

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 [[Place:Western Europe|western Europe]], a similar method was later described by Henry Briggs 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 |url=https://works.bepress.com/tjalling_ypma/7/download/}}</ref>

In order to determine sin 1°, al-Kashi discovered the following formula often attributed to [[Biography:François Viète|François Viète]] in the sixteenth 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>

<math>\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,\!</math>

===''The Key to Arithmetic''===

====Computation of 2{{pi}}====
In his [[Approximations of π|numerical approximation]], he correctly computed 2[[Pi|{{pi}}]] (or <math>\tau</math>) to 9 [[Sexagesimal|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 estimate of 2{{pi}} to 16 [[Decimal|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|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|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 [[Biography:Ptolemy|Ptolemy]], 150 CE), Chinese mathematics (7 decimal places by [[Biography:Zu Chongzhi|Zu Chongzhi]], 480 AD) or Indian mathematics (11 decimal places by [[Biography:Madhava of Sangamagrama|Madhava]] of [[Organization:Kerala School of Astronomy and Mathematics|Kerala School]], ''c.'' 14th Century ). The accuracy of al-Kashi's estimate was not surpassed until [[Biography:Ludolph van Ceulen|Ludolph van Ceulen]] computed 20 decimal places of {{pi}} 180 years later.<ref name=MacTutor>{{MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi}}</ref> Al-Kashi's goal was to compute the circle constant so precisely that the circumference of the largest possible circle (ecliptica) could be computed with the highest desirable precision (the diameter of a hair).

====Decimal fractions====
In discussing decimal fractions, [[Biography:Dirk Jan Struik|Struik]] states that (p.&nbsp;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>
<blockquote>
"The introduction of decimal fractions as a common computational practice can be dated back to the Flemish pamphlet ''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 Northern Netherlands. 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 and [[Sexagesimal|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>"
</blockquote>

====Khayyam's triangle====
In considering [[Pascal's triangle]], known in Persia as "Khayyam's triangle" (named after Omar Khayyám), Struik notes that (p.&nbsp;21):<ref name="Struik1986"/>
<blockquote>
"The Pascal triangle appears for the first time (so far as we know at present) in a book of 1261 written by [[Biography:Yang Hui|Yang Hui]], one of the mathematicians of the Song dynasty in {{wipe|China}}.<ref>J. Needham, ''Science and civilisation in China'', III (Cambridge University Press, New York, 1959), 135.</ref> The properties of [[Binomial coefficient|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 [[Philosophy:Renaissance|Renaissance]] mathematicians, and we see [[Biography:Blaise Pascal|Pascal]]'s triangle on the title page of Peter Apian's [[Social: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>"
</blockquote>

==Biographical film==
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 [[Social:Persian language|Persian]] translation of the title ''Soll'am-os-Samā' '' (سُلّمُ السَماء) of a scientific work by Jamshid Kashani written in [[Social: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 [[Astronomy:Earth|Earth]], the [[Astronomy:Sun|Sun]], the [[Astronomy:Moon|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] {{Webarchive|url=https://web.archive.org/web/20090826104443/http://ch1.iribtv.ir/index.php?option=com_content&task=view&id=5246&Itemid=265 |date=2009-08-26 }}. 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, {{cite web |url=http://www.mehrnews.ir/NewsPrint.aspx?NewsID=808056 |title=Archived copy |accessdate=2009-10-04 |url-status=dead |archiveurl=https://web.archive.org/web/20110722020726/http://www.mehrnews.ir/NewsPrint.aspx?NewsID=808056 |archivedate=2011-07-22 }}.</ref>

==Notes==
{{reflist|2}}

==See also==
*[[Approximations of π|Numerical approximations of {{pi}}]]

==References==
*{{Citation |last=Kennedy |first=Edward S. |year=1947 |title=Al-Kashi's Plate of Conjunctions |journal=Isis |volume=38 |issue=1–2 |pages=56–59 |doi=10.1086/348036}}
*{{Citation |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 |doi=10.1086/349146 |pmid=15436217}}
*{{Citation |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 |doi=10.2307/595221 |publisher=American Oriental Society |jstor=595221}}
*{{Citation |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 |doi=10.1086/349363}}
*{{MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi}}

==External links==
* {{cite encyclopedia |editor=Thomas Hockey |display-editors=etal |last=Schmidl |first=Petra G. |title=Kāshī: Ghiyāth (al‐Milla wa‐) al‐Dīn Jamshīd ibn Masʿūd ibn Maḥmūd al‐Kāshī &#91;al‐Kāshānī&#93; |encyclopedia=The Biographical Encyclopedia of Astronomers |publisher=Springer |date=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])
*[https://web.archive.org/web/20070208101609/http://www.math-cs.cmsu.edu/~mjms/2000.2/azar5.ps Mohammad K. Azarian, A summary of "Miftah al-Hisab", Missouri Journal of Mathematical Sciences, Vol. 12, No. 2, Spring 2000, pp. 75-95]
*[http://www.iranchamber.com/personalities/jkashani/jamshid_kashani.php About Jamshid Kashani]
*[http://www.jphogendijk.nl/arabsci/kashi.html Sources relating to Ghiyath al-Din Kashani, or al-Kashi, by Jan Hogendijk]
{{Commons|Jamshīd al-Kāshī}}
* {{cite journal |url=https://ijpam.eu/contents/2004-14-4/5/5.pdf |journal=International Journal of Pure and Applied Mathematics |title=Al-Kashi's Fundamental Theorem|first1=Mohammad K. |last1=Azarian|year=2004}}
* {{cite journal |url=http://forumgeom.fau.edu/FG2015volume15/FG201523.pdf |journal=Forum Geometricorum |title=A Study of Risa-la al-Watar wa'l Jaib ("The Treatise on the Chord and Sine")|first1=Mohammad K. |last1=Azarian|year=2015}}


{{Islamic astronomy}}
{{Islamic mathematics}}

{{DEFAULTSORT:Kashi, Jamshid}}
[[Category:14th-century mathematicians]]
[[Category:15th-century mathematicians]]

{{Sourceattribution|Jamshīd al-Kāshī|1}}