View source for Jamshid al-Kashi

{{short description|Persian astronomer and mathematician (c. 1380–1429)}}
{{Infobox religious biography
| religion = [[Islam]]
| occupation = [[Persian peoples|Persian]] [[Muslim]] [[scholar]]
| era = [[Islamic Golden Age]]-[[Timurid Renaissance]]
| image = Jamshid bin Mas'ud bin Mahmud al-Tabib al-Kashi, known as Ghiyath (d. 1429); Miftah al-Hisab signed ibn Muhammad Mu'min Taj al-Din al-Shirazi, Iran, dated 31 May 1656.jpg
| caption = Opening bifolio of a manuscript of al-Kashi's ''Miftah al-Hisab''. Copy created in [[Safavid Iran]], dated 1656
| name = Ghiyāth al-Dīn Jamshīd Kāshānī
| title = al-Kashi
| birth_date = {{circa|1380}}
| birth_place = [[Kashan]], [[Iran]]
| death_date = {{death-date and age|22 June 1429|1380}}
| death_place = [[Samarkand]], [[Transoxiana]]
| region = [[Iran]]
| Maddhab = 
| school_tradition = 
| main_interests = [[Islamic astronomy|Astronomy]], [[Islamic mathematics|Mathematics]]
| notable_ideas = [[Pi]] decimal determination to the 16th place<br>[[Law of cosines]]
| works = [[Sullam al-sama']]
| influences =
| influenced =[[Ali Qushji]]<ref name="st-andrews"/>
}}

'''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> ({{langx|fa|غیاث‌الدین جمشید کاشانی}} ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380 [[Kashan]], [[Iran]] – 22 June 1429 [[Samarkand]], [[Transoxiana]]) was a Persian [[Astronomy in the medieval Islamic world|astronomer]] and [[Mathematics in medieval Islam|mathematician]] during the reign of [[Tamerlane]].

Much of al-Kāshī's work was not brought to [[Europe]] and still, even the extant work, remains unpublished in any form.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:Jamshid bin Mas'ud bin Mahmud known as Ghiyath (d. 1429); al-Risala al-Kamaliya, Safavid Iran, dated 26 June 1520.jpg|thumb|Manuscript of al-Kashi's ''al-Risala al-Kamaliya''. Copy created in [[Safavid Iran]], dated 26 June 1520]]
[[File:Resaleye mohitiye end.jpg|thumb|Last page of a copy of ''The Key to Arithmetic'']]

Al-Kashi was born in 1380, in [[Kashan]], in central Iran, to a [[Persian people|Persian]] family.<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> This region was controlled by [[Timur|Tamerlane]], better known as Timur.

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 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, [[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 gathered many great mathematicians and scientists of the [[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, in 1429. Some state that he was murdered and say that Ulugh Beg probably ordered this, whereas others suggest he died a natural death.<ref>{{cite web|title=Jamshid al-Kashi|url=https://www.scientificlib.com/en/Mathematics/Biographies/JamshidAlKashi.html|website=Scientific Lib|access-date=14 November 2023}}</ref><ref>{{cite encyclopedia |title=al-Kāshī |encyclopedia=Encyclopedia Britannica |date=18 June 2023 |last=Dold-Samplonius |first=Yvonne |publisher= |location= |id= |url=https://www.britannica.com/biography/al-Kashi |access-date=14 November 2023 }}</ref> Regardless, after his death, Ulugh Beg described him as "a remarkable scientist" who "could solve the most difficult problems".<ref name="st-andrews">{{cite web|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi|url=https://mathshistory.st-andrews.ac.uk/Biographies/Al-Kashi/|website=Maths History|publisher=[[University of St Andrews]]|last1=O'Connor|first1=J.|last2=Robertson|first2=E.|date=July 1999|year=1999|access-date=14 November 2023}}</ref><ref>B A Rosenfeld, A P Youschkevitch, Biography in Dictionary of Scientific Biography (New York 1970-1990).</ref>

==Astronomy==

===''Khaqani Zij''===
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|theology]]. 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/>

===''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 [[Astronomical object|heavenly bodies]], such as the [[Earth]], the [[Moon]], the [[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 [[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|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 conjunction]]s will occur,<ref>{{Harv|Kennedy|1947|p=56}}</ref> and for performing [[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 [[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>

==Mathematics==


===Computation of {{math|2''π''}} ===
Al-Kashi made the most accurate [[approximations of π|approximation of {{math|''π''}}]] to date in his ''al-Risāla al-muhītīyya'' (''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|doi=10.22052/mir.2019.167225.1110}}</ref> He correctly computed [[pi|{{math|2''π''}}]] 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 estimate of 2{{pi}} to 16 [[decimal]] places of accuracy.<ref>The statement that a quantity is calculated to <math>n</math> sexagesimal digits implies that the maximal ''inaccuracy'' in the calculated value is less than  <math>\textstyle 59/60^{n+1} + 59/60^{n+2} + \dots = 1/60^n</math> in the [[decimal]] system. With <math>n=9</math>, Al-Kashi has thus calculated <math>2\pi</math> with a maximal ''error'' less than <math>\textstyle 1/60^{9}\approx 9.92\times 10^{-17} < 10^{-16}\,</math>. That is to say, Al-Kashi has calculated <math>2\pi</math> exactly up to and including the 16th place after the [[decimal separator]]. For <math>2\pi</math> expressed exactly up to and including the 18th place after the decimal separator one has: <math> 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 [[Ptolemy]], AD 150), [[Chinese mathematics]] (7 decimal places by [[Zu Chongzhi]], AD 480) or [[Indian mathematics]] (11 decimal places by [[Madhava of Sangamagrama|Madhava]] of [[Kerala School of Astronomy and Mathematics|Kerala School]], ''c.'' 14th Century). The accuracy of al-Kashi's estimate was not surpassed until [[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).

===''Treatise on the Chord and Sine''===
In Al-Kashi's ''Risālah al-watar waʾl-jaib'' (''Treatise on the Chord and Sine''), he 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 ad-Din Muhammad ibn Ma'ruf|Taqi al-Din]] in the sixteenth 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/>

A method algebraically equivalent to [[Newton's method]] was known to his predecessor [[Sharaf al-Din al-Tusi]]. 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 Briggs (mathematician)|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 [[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''===

====Law of cosines====
[[File:Law of cosines following al-Kashi.png|thumb|right|upright=1.25|Al-Kashi's version of the law of cosines (case where {{mvar|γ}} is obtuse), expressed with modern algebraic notation.]]
Al-Kashi's ''Miftāḥ al-ḥisāb'' (''Key of Arithmetic'', 1427) explained how to [[solution of triangles|solve triangles]] from various combinations of given data. The method used when two sides and their included angle were given was essentially the same method used by 13th century Persian mathematician [[Nasir al-Din al-Tusi|Naṣīr al-Dīn al-Ṭūsī]] in his {{lang|ar|Kitāb al-Shakl al-qattāʴ}} (''Book on the Complete Quadrilateral'', c. 1250),<ref>{{cite book |title=Traité du quadrilatère attribué a Nassiruddinel-Toussy |chapter=Ch. 3.2: Sur la manière de calculer les côtés et les angles d'un triangle les uns par les autres |page=69 |language=fr |author=Naṣīr al-Dīn al-Ṭūsī |translator-last=Caratheodory |translator-first=Alexandre Pacha |translator-link=Alexander Karatheodori Pasha |chapter-url=https://books.google.com/books?id=gVE7AQAAIAAJ&pg=PA69 |publisher=Typographie et Lithographie Osmanié |year=1891 |quote=On donne deux côtés et un angle. [...] Que si l'angle donné est compris entre les deux côtés donnés, comme l'angle A est compris entre les deux côtés AB AC, abaissez de B sur AC la perpendiculaire BE. Vous aurez ainsi le triangle rectangle [BEA] dont nous connaissons le côté AB et l'angle A; on en tirera BE, EA, et l'on retombera ainsi dans un des cas précédents; c. à. d. dans le cas où BE, CE sont connus; on connaîtra dès lors BC et l'angle C, comme nous l'avons expliqué |trans-quote=Given [...] the angle A is included between the two sides AB AC, drop from B to AC the perpendicular BE. You will thus have the right triangle [BEA] of which we know the side AB and the angle A; in that triangle compute BE, EA, and the problem is reduced to one of the preceding cases; that is, to the case where BE, CE are known; we will thus know BC and the angle C, as we have explained.}}</ref> but Al-Kashi presented all of the steps instead of leaving details to the reader:

{{quote|text='''Another case''' is when two sides and the angle between them are known and the rest are unknown. We multiply one of the sides by the sine of the [known] angle one time and by the sine of its complement the other time converted and we subtract the second result from the other side if the angle is acute and add it if the angle is obtuse. We then square the result and add to it the square of the first result. We take the square root of the sum to get the remaining side....|sign=Al-Kāshī's {{lang|ar|Miftāḥ al-ḥisāb}},<br>{{in5}}translation by Nuh Aydin, Lakhdar Hammoudi, and Ghada Bakbouk<ref>{{cite book |last1=Aydin |first1=Nuh |last2=Hammoudi |first2=Lakhdar |last3=Bakbouk |first3=Ghada |year=2020 |page=31 |title=Al-Kashi's Miftah al-Hisab, Volume II: Geometry |publisher=Birkhäuser |doi=10.1007/978-3-030-61330-3|isbn=978-3-030-61329-7 }}</ref> }}

Using modern algebraic notation and conventions this might be written

:<math>c = \sqrt{(b - a \cos \gamma)^2 + (a \sin \gamma)^2}</math>

After applying the Pythagorean trigonometric identity <math>\cos^2\gamma + \sin^2\gamma = 1,</math> this is algebraically equivalent to the modern [[law of cosines]]:

:<math>\begin{align}
c^2
&= b^2 - 2ba\cos \gamma + a^2\cos^2 \gamma + a^2\sin^2\gamma \\[5mu]
&= a^2 + b^2 - 2ab\cos \gamma.
\end{align}</math>

In [[France]], the law of cosines is sometimes referred to as the ''théorème d'Al-Kashi''.<ref name = Pickover>{{Cite book
|url=https://books.google.com/books?id=JrslMKTgSZwC&q=al+kashi+law+of+cosines&pg=PA106
|title=The Math Book: From Pythagoras to the 57th Dimension |last=Pickover |first=Clifford A. |date=2009 |publisher=Sterling Publishing Company, Inc. |isbn=9781402757969 |page=106}}</ref><ref>{{Cite book |title=Programme de mathématiques de première générale |publisher=Ministère de l'Éducation nationale et de la Jeunesse |year=2022 |pages=11,12 |language=fr}}</ref>

====Decimal fractions====
In discussing [[decimal fractions]], [[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 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>"
</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 [[Yang Hui]], one of the mathematicians of the [[Song 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>"
</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 [[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, with each part being 45 minutes long, 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 |access-date=2009-10-04 |url-status=dead |archive-url=https://web.archive.org/web/20110722020726/http://www.mehrnews.ir/NewsPrint.aspx?NewsID=808056 |archive-date=2011-07-22 }}.</ref>

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

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

==References==
*{{Citation |last=Kennedy |first=Edward S. |authorlink=Edward Stewart Kennedy|year=1947 |title=Al-Kashi's Plate of Conjunctions |journal=[[Isis (journal)|Isis]] |volume=38 |issue=1–2 |pages=56–59 |doi=10.1086/348036|s2cid=143993402 }}
*{{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 (journal)|Isis]] |volume=41 |issue=2 |pages=180–183 |doi=10.1086/349146 |pmid=15436217|s2cid=43217299 }}
*{{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 (journal)|Isis]] |volume=43 |issue=1 |pages=42–50 |doi=10.1086/349363|s2cid=123582209 }}
*{{MacTutor|id=Al-Kashi|title=Ghiyath al-Din Jamshid Mas'ud al-Kashi}}

==External links==
{{Commons|Jamshīd al-Kāshī}}
* {{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])
* {{cite journal |last1=Eshera |first1=Osama |title=On the Early Collections of the Works of Ġiyāṯ al-Dīn Jamšīd al-Kāšī |journal=Journal of Islamic Manuscripts |date=2020 |volume=13 |issue=2 |pages=225–262 |doi=10.1163/1878464X-01302001|s2cid=248336832 }}
*[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]
* {{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}}
* {{cite journal |url=http://forumgeom.fau.edu/FG2018volume18/FG201828.pdf |journal=Forum Geometricorum |title=A Study of Risa-la al-Watar wa'l Jaib ("The Treatise on the Chord and Sine"):Revisited|first1=Mohammad K. |last1=Azarian|year=2018}}
* {{cite journal |url=https://ijpam.eu/contents/2009-57-6/5/5.pdf |journal=International Journal of Pure and Applied Mathematics |title=The Introduction of Al-Risala al-Muhitiyya: An English Translation|first1=Mohammad K. |last1=Azarian|year=2009}}

{{Mathematics in Iran}}
{{Islamic astronomy}}
{{Islamic mathematics}}
{{Authority control}}

{{DEFAULTSORT:Kashi, Jamshid}}
[[Category:1380s births]]
[[Category:1429 deaths]]
[[Category:People from Kashan]]
[[Category:15th-century Iranian mathematicians]]
[[Category:Medieval Iranian astrologers]]
[[Category:15th-century Iranian astronomers]]
[[Category:15th-century astrologers]]
[[Category:Medieval Iranian physicists]]
[[Category:Scholars from the Timurid Empire]]
[[Category:15th-century inventors]]