Jamshīd alKāshī
Ghiyāth alDīn Jamshīd Kāshānī  

Opening bifolio of a manuscript of alKashi's Miftah alHisab. Copy created in Safavid Iran, dated 1656  
Title  alKashi 
Personal  
Born  c. 1380 Kashan, Iran 
Died  22 June 1429 (14290623) (aged 48) Samarkand, Transoxania 
Religion  Islam 
Era  Islamic Golden AgeTimurid Renaissance 
Region  Iran 
Main interest(s)  Astronomy, Mathematics 
Notable idea(s)  Pi decimal determination to the 16th place Law of cosines 
Notable work(s)  Sullam alSama 
Occupation  Persian Muslim scholar 
Ghiyāth alDīn Jamshīd Masʿūd alKāshī (or alKāshānī)^{[1]} (Persian: غیاث الدین جمشید کاشانی Ghiyāsuddīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician during the reign of Tamerlane.^{[2]}^{[3]}
Much of alKāshī's work was not brought to Europe, and still, even the extant work, remains unpublished in any form.^{[4]}
Biography
AlKashi 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 alKashi 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, alKashi took this opportunity to contribute vast amounts of knowledge to his people. His best work was done in the court of Ulugh Beg.
AlKashi was still working on his book, called “Risala alwatar wa’ljaib” 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.^{[citation needed]}
Astronomy
Khaqani Zij
AlKashi produced a Zij entitled the Khaqani Zij, which was based on Nasir alDin alTusi's earlier Ziji Ilkhani. In his Khaqani Zij, alKashi thanks the Timurid sultan and mathematicianastronomer Ulugh Beg, who invited alKashi to work at his observatory (see Islamic astronomy) and his university (see Madrasah) which taught theology. AlKashi produced 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 systems on the celestial sphere, such as the transformation from the ecliptic coordinate system to the equatorial coordinate system.^{[5]}
Astronomical Treatise on the size and distance of heavenly bodies
He wrote the book Sullam alSama on the resolution of difficulties met by predecessors in the determination of distances and sizes of heavenly bodies, such as the Earth, the Moon, the Sun, and the Stars.
Treatise on Astronomical Observational Instruments
In 1416, alKashi wrote the Treatise on Astronomical Observational Instruments, which described a variety of different instruments, including the triquetrum and armillary sphere, the equinoctial armillary and solsticial armillary of Mo'ayyeduddin Urdi, the sine and versine instrument of Urdi, the sextant of alKhujandi, the Fakhri sextant at the Samarqand observatory, a double quadrant Azimuthaltitude instrument he invented, and a small armillary sphere incorporating an alhidade which he invented.^{[6]}
Plate of Conjunctions
AlKashi invented the Plate of Conjunctions, an analog computing instrument used to determine the time of day at which planetary conjunctions will occur,^{[7]} and for performing linear interpolation.^{[8]}
Planetary computer
AlKashi 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 the Sun and Moon,^{[8]} and the planets in terms of elliptical orbits;^{[9]} the latitudes of the Sun, Moon, and planets; and the ecliptic of the Sun. The instrument also incorporated an alhidade and ruler.^{[10]}
Mathematics
Law of cosines
In French, the law of cosines is named Théorème d'AlKashi (Theorem of AlKashi), as alKashi was the first to provide an explicit statement of the law of cosines in a form suitable for triangulation.^{[11]} His other work is alRisāla almuhītīyya or "The Treatise on the Circumference".^{[12]}
The Treatise of Chord and Sine
In The Treatise on the Chord and Sine, alKashi 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 alDin in the sixteenth century. In algebra and numerical analysis, he developed an iterative method for solving cubic equations, which was not discovered in Europe until centuries later.^{[5]}
A method algebraically equivalent to Newton's method was known to his predecessor Sharaf alDīn alTūsī. AlKāshī improved on this by using a form of Newton's method to solve $x^{P}N=0$ to find roots of N. In western Europe, a similar method was later described by Henry Briggs in his Trigonometria Britannica, published in 1633.^{[13]}
In order to determine sin 1°, alKashi discovered the following formula, often attributed to François Viète in the sixteenth century:^{[14]}
$\sin 3\phi =3\sin \phi 4\sin ^{3}\phi \,\!$
The Key to Arithmetic
Computation of 2π
In his numerical approximation, he correctly computed 2π to 9 sexagesimal digits^{[15]} in 1424,^{[5]} and he converted this estimate of 2π to 16 decimal places of accuracy.^{[16]} 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 Kerala School, c. 14th Century ). The accuracy of alKashi's estimate was not surpassed until Ludolph van Ceulen computed 20 decimal places of π 180 years later.^{[5]} AlKashi'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, Struik states that (p. 7):^{[17]}
"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 (15481620), 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 AlKāshī used both decimal and sexagesimal fractions with great ease in his Key to arithmetic (Samarkand, early fifteenth century).^{[18]}"
Khayyam's triangle
In considering Pascal's triangle, known in Persia as "Khayyam's triangle" (named after Omar Khayyám), Struik notes that (p. 21):^{[17]}
"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.^{[19]} The properties of binomial coefficients were discussed by the Persian mathematician Jamshid AlKāshī in his Key to arithmetic of c. 1425.^{[20]} 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 Pascal's triangle on the title page of Peter Apian's German arithmetic of 1527. After this, we find the triangle and the properties of binomial coefficients in several other authors.^{[21]}"
Biographical film
In 2009, IRIB produced and broadcast (through Channel 1 of IRIB) a biographicalhistorical film series on the life and times of Jamshid AlKāshi, with the title The Ladder of the Sky ^{[22]}^{[23]} (Nardebāme Āsmān ^{[24]}). The series, which consists of 15 parts, with each part being 45 minutes long, is directed by Mohammad Hossein Latifi and produced by Mohsen AliAkbari. In this production, the role of the adult Jamshid AlKāshi is played by Vahid Jalilvand.^{[25]}^{[26]}^{[27]}
Notes
 ^ A. P. Youschkevitch and B. A. Rosenfeld. "alKāshī (alKāshānī), Ghiyāth alDīn Jamshīd Masʿūd" Dictionary of Scientific Biography.
 ^ Bosworth, C.E. (1990). The Encyclopaedia of Islam, Volume IV (2. impression. ed.). Leiden [u.a.]: Brill. p. 702. ISBN 9004057455.
ALKASHl Or ALKASHANI, GHIYATH ALDIN DjAMSHlD B. MASCUD B. MAHMUD, Persian mathematician and astronomer who wrote in his mother tongue and in Arabic.
 ^ Selin, Helaine (2008). Encyclopaedia of the history of science, technology, and medicine in nonwestern cultures. Berlin New York: Springer. p. 132. ISBN 9781402049606.
AlKāshī, or alKāshānī (Ghiyāth alDīn Jamshīd ibn Mas˓ūd alKāshī (alKāshānī)), was a Persian mathematician and astronomer.
 ^ [1] iranicaonline.org
 ^ ^{a} ^{b} ^{c} ^{d} O'Connor, John J.; Robertson, Edmund F., "Ghiyath alDin Jamshid Mas'ud alKashi", MacTutor History of Mathematics archive, University of St Andrews
 ^ (Kennedy 1951, pp. 104–107)
 ^ (Kennedy 1947, p. 56)
 ^ ^{a} ^{b} (Kennedy 1950)
 ^ (Kennedy 1952)
 ^ (Kennedy 1951)
 ^ Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling Publishing Company, Inc. p. 106. ISBN 9781402757969.
 ^ Azarian, Mohammad K. (2019). "An Overview of Mathematical Contributions of Ghiyath alDin Jamshid AlKashi [Kashani]" (PDF). Mathematics Interdisciplinary Research. 4 (1). doi:10.22052/mir.2019.167225.1110.
 ^ Ypma, Tjalling J. (December 1995), "Historical Development of the NewtonRaphson Method", SIAM Review, Society for Industrial and Applied Mathematics, 37 (4): 531–551 [539], doi:10.1137/1037125
 ^ Marlow Anderson, Victor J. Katz, Robin J. Wilson (2004), Sherlock Holmes in Babylon and Other Tales of Mathematical History, Mathematical Association of America, p. 139, ISBN 0883855461
 ^ AlKashi, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256
 ^ The statement that a quantity is calculated to $\scriptstyle n$ sexagesimal digits implies that the maximal inaccuracy in the calculated value is less than $\scriptstyle 59/60^{n+1}+59/60^{n+2}+\dots =1/60^{n}$ in the decimal system. With $\scriptstyle n=9$, AlKashi has thus calculated $\scriptstyle 2\pi$ with a maximal error less than $\scriptstyle 1/60^{9}\approx 9.92\times 10^{17}<10^{16}\,$. That is to say, AlKashi has calculated $\scriptstyle 2\pi$ exactly up to and including the 16th place after the decimal separator. For $\scriptstyle 2\pi$ expressed exactly up to and including the 18th place after the decimal separator one has: $\scriptstyle 6.283\,185\,307\,179\,586\,476$.
 ^ ^{a} ^{b} D.J. Struik, A Source Book in Mathematics 12001800 (Princeton University Press, New Jersey, 1986). ISBN 0691023972
 ^ P. Luckey, Die Rechenkunst bei Ğamšīd b. Mas'ūd alKāšī (Steiner, Wiesbaden, 1951).
 ^ J. Needham, Science and civilisation in China, III (Cambridge University Press, New York, 1959), 135.
 ^ Russian translation by B.A. Rozenfel'd (Gos. Izdat, Moscow, 1956); see also Selection I.3, footnote 1.
 ^ Smith, History of mathematics, II, 508512. See also our Selection II.9 (Girard).
 ^ 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, [2].
 ^ IRIB to spice up Ramadan evenings with special series, Tehran Times, 22 August 2009, [3].
 ^ The name Nardebāme Āsmān coincides with the Persian translation of the title Soll'amosSamā' (سُلّمُ السَماء) of a scientific work by Jamshid Kashani written in Arabic. In this work, which is also known as Resālehye 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.
 ^ The programmes of the Holy month of Ramadan, Channel 1, in Persian, 19 August 2009, [4] Archived 20090826 at the Wayback Machine. Here the name "Latifi" is incorrectly written as "Seifi".
 ^ Dr Velāyati: 'The Ladder of the Sky' is faithful to history, in Persian, Āftāb, Tuesday, 1 September 2009, [5].
 ^ 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, "Archived copy". Archived from the original on 20110722. Retrieved 20091004.
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: CS1 maint: archived copy as title (link).
See also
 Numerical approximations of π
References
 Kennedy, Edward S. (1947), "AlKashi's Plate of Conjunctions", Isis, 38 (1–2): 56–59, doi:10.1086/348036, S2CID 143993402
 Kennedy, Edward S. (1950), "A FifteenthCentury Planetary Computer: alKashi's "Tabaq alManateq" I. Motion of the Sun and Moon in Longitude", Isis, 41 (2): 180–183, doi:10.1086/349146, PMID 15436217, S2CID 43217299
 Kennedy, Edward S. (1951), "An Islamic Computer for Planetary Latitudes", Journal of the American Oriental Society, American Oriental Society, 71 (1): 13–21, doi:10.2307/595221, JSTOR 595221
 Kennedy, Edward S. (1952), "A FifteenthCentury Planetary Computer: alKashi's "Tabaq alManeteq" II: Longitudes, Distances, and Equations of the Planets", Isis, 43 (1): 42–50, doi:10.1086/349363, S2CID 123582209
 O'Connor, John J.; Robertson, Edmund F., "Ghiyath alDin Jamshid Mas'ud alKashi", MacTutor History of Mathematics archive, University of St Andrews
External links
 Schmidl, Petra G. (2007). "Kāshī: Ghiyāth (al‐Milla wa‐) al‐Dīn Jamshīd ibn Masʿūd ibn Maḥmūd al‐Kāshī [al‐Kāshānī]". In Thomas Hockey; et al. (eds.). The Biographical Encyclopedia of Astronomers. New York: Springer. pp. 613–5. ISBN 9780387310220. (PDF version)
 Eshera, Osama (2020). "On the Early Collections of the Works of Ġiyāṯ alDīn Jamšīd alKāšī". Journal of Islamic Manuscripts. 13 (2): 225–262. doi:10.1163/1878464X01302001. S2CID 248336832.
 Mohammad K. Azarian, A summary of "Miftah alHisab", Missouri Journal of Mathematical Sciences, Vol. 12, No. 2, Spring 2000, pp. 7595
 About Jamshid Kashani
 Sources relating to Ghiyath alDin Kashani, or alKashi, by Jan Hogendijk
 Azarian, Mohammad K. (2004). "AlKashi's Fundamental Theorem" (PDF). International Journal of Pure and Applied Mathematics.
 Azarian, Mohammad K. (2015). "A Study of Risala alWatar wa'l Jaib ("The Treatise on the Chord and Sine")" (PDF). Forum Geometricorum.
 Azarian, Mohammad K. (2018). "A Study of Risala alWatar wa'l Jaib ("The Treatise on the Chord and Sine"):Revisited" (PDF). Forum Geometricorum.
 Azarian, Mohammad K. (2009). "The Introduction of AlRisala alMuhitiyya: An English Translation" (PDF). International Journal of Pure and Applied Mathematics.
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