# Jamshīd al-Kāshī

**Ghiyāth al-Dīn Jamshīd ibn Masʾūd al-Kāshī** (or **Jamshīd Kāshānī**, Persian: غیاثالدین جمشید کاشانی) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persian astronomer and mathematician.

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## Biography

Al-Kashi was one of the best mathematicians in the Islamic world. He was born in 1380, in Kashan, 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 the sciences, 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.

Eight years after he came into power in 1409, Ulugh Beg founded an institute in Samarkand which soon became the world’s most prestigious 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.

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.

## 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 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 Islamic theology as well as Islamic science. Al-Kashi 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.^{[1]}

### *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 and armillary sphere, the equinoctial armillary and solsticial armillary of Mo'ayyeduddin Urdi, the sine and versine instrument of Urdi, the 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.^{[2]}

#### Plate of Conjunctions

Al-Kashi invented the Plate of Conjunctions, an analog computing instrument used to determine the time of day at which planetary conjunctions will occur,^{[3]} and for performing linear interpolation.^{[4]}

#### 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 the Sun and Moon,^{[4]} and the planets in terms of elliptical orbits;^{[5]} the latitudes of the Sun, Moon, and planets; and the ecliptic of the Sun. The instrument also incorporated an alhidade and ruler.^{[6]}

## 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 for triangulation.

### *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. In algebra and numerical analysis, he developed an iterative method for solving cubic equations, which was not discovered in Europe until centuries later.^{[1]}

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 *x*^{P} − *N* = 0 to find roots of *N*. In western Europe, a similar method was later described by Henry Biggs in his *Trigonometria Britannica*, published in 1633.^{[7]}

In order to determine sin 1°, al-Kashi discovered the following formula often attributed to François Viète in the 16th century:^{[8]}

### *The Key to Arithmetic*

#### Computation of π

In one of his numerical approximations of π, he correctly computed 2π to 9 sexagesimal digits.^{[9]} This approximation of 2π is equivalent to 16 decimal places of accuracy.^{[10]} 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.^{[1]}

#### Decimal fractions

In discussing decimal fractions, Struik states that (p. 7):^{[11]}

"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 fractions with great ease in hisKey to arithmetic(Samarkand, early fifteenth century).^{[12]}"

#### Khayyam's triangle

In considering Pascal's triangle, known in Persia as "Khayyam's triangle" (named after Omar Khayyám), Struik notes that (p. 21):^{[11]}

"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.

^{[13]}The properties of binomial coefficients were discussed by the Persian mathematician Jamshid Al-Kāshī in hisKey to arithmeticof c. 1425.^{[14]}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.^{[15]}"

## Notes

- ^
^{a}^{b}^{c}O'Connor, John J.; Robertson, Edmund F., "Ghiyath al-Din Jamshid Mas'ud al-Kashi",*MacTutor History of Mathematics archive*. **^**(Kennedy 1961, pp. 104-107)**^**(Kennedy 1947, p. 56)- ^
^{a}^{b}(Kennedy 1950) **^**(Kennedy 1952)**^**(Kennedy 1951)**^**Ypma, Tjalling J. (December 1995), "Historical Development of the Newton-Raphson Method",*SIAM Review*(Society for Industrial and Applied Mathematics)**37**(4): 531–551 [539], doi:**^**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**^***Al-Kashi*, author: Adolf P. Youschkevitch, chief editor: Boris A. Rosenfeld, p. 256**^**The statement that a quantity is calculated to sexagesimal digits implies that the maximal*inaccuracy*in the calculated value is less than in the decimal system. With , Al-Kashi has thus calculated with a maximal*error*less than . That is to say, Al-Kashi has calculated exactly up to and including the 16th place after the decimal separator. For expressed exactly up to and including the 18th place after the decimal separator one has: .- ^
^{a}^{b}D.J. Struik,*A Source Book in Mathematics 1200-1800*(Princeton University Press, New Jersey, 1986). ISBN 0-691-02397-2 **^**P. Luckey,*Die Rechenkunst bei Ğamšīd b. Mas'ūd al-Kāšī*(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, 508-512. See also our Selection**II.9**(Girard).

## Related

- History of numerical approximations of π

## References

- Kennedy, Edward S. (1947), "Al-Kashi's Plate of Conjunctions",
*Isis***38**(1-2): 56-59 - Kennedy, Edward S. (1950), "A Fifteenth-Century Planetary Computer: al-Kashi's "Tabaq al-Manateq" I. Motion of the Sun and Moon in Longitude",
*Isis***41**(2): 180-183 - Kennedy, Edward S. (1951), "An Islamic Computer for Planetary Latitudes",
*Journal of the American Oriental Society***71**(1): 13-21 - Kennedy, Edward S. (1952), "A Fifteenth-Century Planetary Computer: al-Kashi's "Tabaq al-Maneteq" II: Longitudes, Distances, and Equations of the Planets",
*Isis***43**(1): 42-50 - O'Connor, John J.; Robertson, Edmund F., "Ghiyath al-Din Jamshid Mas'ud al-Kashi",
*MacTutor History of Mathematics archive*.