Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) ( fa, غیاث الدین جمشید کاشانی ''Ghiyās-ud-dīn Jamshīd Kāshānī'') (c. 1380

Kashan Kashan ( fa, ; Qashan; Cassan; also romanized as Kāshān) is a city in the northern part of Isfahan province, Iran. At the 2017 census, its population was 396,987 in 90,828 families. Some etymologists argue that the city name comes from ...

Iran Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkmeni ...

– 22 June 1429 Samarkand,

Transoxania Transoxiana or Transoxania (Land beyond the Oxus) is the Latin name for a region and civilization located in lower Central Asia roughly corresponding to modern-day eastern Uzbekistan, western Tajikistan, parts of southern Kazakhstan, parts of ...

) was a

Persian Persian may refer to: * People and things from Iran, historically called ''Persia'' in the English language ** Persians, the majority ethnic group in Iran, not to be conflated with the Iranic peoples ** Persian language, an Iranian language of the ...

astronomer An astronomer is a scientist in the field of astronomy who focuses their studies on a specific question or field outside the scope of Earth. They observe astronomical objects such as stars, planets, moons, comets and galaxies – in either ...

and

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

during the reign of

Tamerlane Timur ; chg, ''Aqsaq Temür'', 'Timur the Lame') or as ''Sahib-i-Qiran'' ( 'Lord of the Auspicious Conjunction'), his epithet. ( chg, ''Temür'', 'Iron'; 9 April 133617–19 February 1405), later Timūr Gurkānī ( chg, ''Temür Kür ...

. Much of al-Kāshī's work was not brought to Europe, and still, even the extant work, remains unpublished in any form.

Biography

Al-Kashi was born in 1380, in

, in central Iran. This region was controlled by

, 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,

Ulugh Beg Mīrzā Muhammad Tāraghay bin Shāhrukh ( chg, میرزا محمد طارق بن شاہ رخ, fa, میرزا محمد تراغای بن شاہ رخ), better known as Ulugh Beg () (22 March 1394 – 27 October 1449), was a Timurid sultan, as ...

, founded an institute in Samarkand which soon became a prominent university. Students from all over the

Middle East The Middle East ( ar, الشرق الأوسط, ISO 233: ) is a geopolitical region commonly encompassing Arabia (including the Arabian Peninsula and Bahrain), Asia Minor (Asian part of Turkey except Hatay Province), East Thrace (Europ ...

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

. 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 '' 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

, who invited al-Kashi to work at his observatory (see

Islamic astronomy Islamic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age (9th–13th centuries), and mostly written in the Arabic language. These developments mostly took place in the Middle ...

) and his

university A university () is an institution of higher (or tertiary) education and research which awards academic degrees in several academic disciplines. Universities typically offer both undergraduate and postgraduate programs. In the United States ...

(see Madrasah) which taught

theology Theology is the systematic study of the nature of the divine and, more broadly, of religious belief. It is taught as an academic discipline, typically in universities and seminaries. It occupies itself with the unique content of analyzing the ...

. Al-Kashi produced sine tables to four

sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...

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 The ecliptic coordinate system is a celestial coordinate system commonly used for representing the apparent positions, orbits, and pole orientations of Solar System objects. Because most planets (except Mercury) and many small Solar System b ...

to the

equatorial coordinate system The equatorial coordinate system is a celestial coordinate system widely used to specify the positions of celestial objects. It may be implemented in spherical or rectangular coordinates, both defined by an origin at the centre of Earth, a fu ...

''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 heavenly bodies, such as the

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ...

, the

Moon The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

, the

Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...

, and the

Stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...

''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 The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',sextant of

al-Khujandi Abu Mahmud Hamid ibn al-Khidr al-Khojandi (known as Abu Mahmood Khojandi, Alkhujandi or al-Khujandi, Persian: ابومحمود خجندی, c. 940 - 1000) was a Muslim Transoxanian astronomer and mathematician born in Khujand (now part of Tajikista ...

, the Fakhri sextant at the

Samarqand fa, سمرقند , native_name_lang = , settlement_type = City , image_skyline = , image_caption = Clockwise from the top: Registan square, Shah-i-Zinda necropolis, Bibi-Khanym Mosque, view inside Shah-i-Zin ...

observatory, a double quadrant

Azimuth An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematical ...

altitude Altitude or height (also sometimes known as depth) is a distance measurement, usually in the vertical or "up" direction, between a reference datum and a point or object. The exact definition and reference datum varies according to the context ...

instrument he invented, and a small armillary sphere incorporating an alhidade which he invented.

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, and for performing

linear interpolation In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known poi ...

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 Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek lette ...

of the

and

, and the

planet A planet is a large, rounded astronomical body that is neither a star nor its remnant. The best available theory of planet formation is the nebular hypothesis, which posits that an interstellar cloud collapses out of a nebula to create a you ...

s in terms of

elliptical orbit In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, i ...

s; the

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

s of the Sun, Moon, and planets; and the

ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...

of the Sun. The instrument also incorporated an alhidade and

ruler A ruler, sometimes called a rule, line gauge, or scale, is a device used in geometry and technical drawing, as well as the engineering and construction industries, to measure distances or draw straight lines. Variants Rulers have long ...

Mathematics

Law of cosines

In French, the

law of cosines In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states ...

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. His other work is al-''Risāla al''-''muhītīyya'' or "The Treatise on the Circumference".

''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 , 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 () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

and

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods ...

, he developed an

iterative method In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived from the pr ...

for solving

cubic equation In algebra, a cubic equation in one variable is an equation of the form :ax^3+bx^2+cx+d=0 in which is nonzero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of th ...

s, which was not discovered in Europe until centuries later. A method algebraically equivalent to Newton's method was known to his predecessor

Sharaf al-Dīn al-Tūsī Sharaf al-Dīn al-Muẓaffar ibn Muḥammad ibn al-Muẓaffar al-Ṭūsī ( fa, شرف‌الدین مظفر بن محمد بن مظفر توسی; 1135 – 1213) was an Iranian mathematician and astronomer of the Islamic Golden Age (during the ...

. 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 Western Europe is the western region of Europe. The region's countries and territories vary depending on context. The concept of "the West" appeared in Europe in juxtaposition to "the East" and originally applied to the ancient Mediterranean ...

, a similar method was later described by

Henry Briggs Henry Briggs may refer to: *Henry Briggs (mathematician) (1561–1630), English mathematician *Henry Perronet Briggs (1793–1844), English painter *Henry George Briggs (1824–1872), English merchant, traveller, and orientalist *Henry Shaw Briggs ...

in his ''Trigonometria Britannica'', published in 1633. In order to determine sin 1°, al-Kashi discovered the following formula, often attributed to

François Viète François Viète, Seigneur de la Bigotière ( la, Franciscus Vieta; 1540 – 23 February 1603), commonly know by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to i ...

in the sixteenth century:

\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,\!

''The Key to Arithmetic''

Computation of 2

In his

numerical approximation Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

, he correctly computed 2 to 9

digits in 1424, and he converted this estimate of 2 to 16 decimal places of accuracy. This was far more accurate than the estimates earlier given in Greek mathematics (3 decimal places by

Ptolemy Claudius Ptolemy (; grc-gre, Πτολεμαῖος, ; la, Claudius Ptolemaeus; AD) was a mathematician, astronomer, astrologer, geographer, and music theorist, who wrote about a dozen scientific treatises, three of which were of importance ...

, AD 150), Chinese mathematics (7 decimal places by

Zu Chongzhi Zu Chongzhi (; 429–500 AD), courtesy name Wenyuan (), was a Chinese astronomer, mathematician, politician, inventor, and writer during the Liu Song and Southern Qi dynasties. He was most notable for calculating pi as between 3.1415926 and 3 ...

, AD 480) or Indian mathematics (11 decimal places by

Madhava Mādhava means Lord Krishna an incarnation of Vishnu. It may also refer to: *a Sanskrit patronymic, "descendant of Madhu (a man of the Yadu tribe)". ** especially of Krishna, see Madhava (Vishnu) *** an icon of Krishna ** Madhava of Sangamagrama, ...

of Kerala School, ''c.'' 14th Century ). The accuracy of al-Kashi's estimate was not surpassed until

Ludolph van Ceulen Ludolph van Ceulen (, ; 28 January 1540 – 31 December 1610) was a German-Dutch mathematician from Hildesheim. He emigrated to the Netherlands. Biography Van Ceulen moved to Delft most likely in 1576 to teach fencing and mathematics and in 159 ...

computed 20 decimal places of 180 years later. 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 The decimal numeral system (also called the base-ten positional numeral system and denary or decanary) is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral ...

, Struik states that (p. 7):D.J. Struik, ''A Source Book in Mathematics 1200-1800'' (Princeton University Press, New Jersey, 1986).

"The introduction of decimal fractions as a common computational practice can be dated back to the
Flemish Flemish (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Southern Dutch (). Flemish is native to Flanders, a historical region in northern Belgium; ...
pamphlet ''De Thiende'', published at
Leyden Leiden (; in English and archaic Dutch also Leyden) is a city and municipality in the province of South Holland, Netherlands. The municipality of Leiden has a population of 119,713, but the city forms one densely connected agglomeration wit ...
in 1585, together with a French translation, ''La Disme'', by the Flemish mathematician
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
(1548-1620), then settled in the Northern
Netherlands ) , anthem = ( en, "William of Nassau") , image_map = , map_caption = , subdivision_type = Sovereign state , subdivision_name = Kingdom of the Netherlands , established_title = Before independence , established_date = Spanish Netherl ...
. It is true that decimal fractions were used by the
Chinese Chinese can refer to: * Something related to China * Chinese people, people of Chinese nationality, citizenship, and/or ethnicity **''Zhonghua minzu'', the supra-ethnic concept of the Chinese nation ** List of ethnic groups in China, people of ...
many centuries before Stevin and that the Persian astronomer Al-Kāshī used both decimal and
sexagesimal Sexagesimal, also known as base 60 or sexagenary, is a numeral system with sixty as its base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified form ...
fractions with great ease in his ''Key to arithmetic'' (Samarkand, early fifteenth century)."

Khayyam's triangle

In considering

Pascal's triangle In mathematics, Pascal's triangle is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although o ...

, known in Persia as "Khayyam's triangle" (named after Omar Khayyám), Struik notes that (p. 21):

"The Pascal triangle appears for the first time (so far as we know at present) in a book of 1261 written by
Yang Hui Yang Hui (, ca. 1238–1298), courtesy name Qianguang (), was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theo ...
, one of the mathematicians of the
Song dynasty The Song dynasty (; ; 960–1279) was an imperial dynasty of China that began in 960 and lasted until 1279. The dynasty was founded by Emperor Taizu of Song following his usurpation of the throne of the Later Zhou. The Song conquered the rest ...
in China. The properties of binomial coefficients were discussed by the Persian mathematician Jamshid Al-Kāshī in his ''Key to arithmetic'' of c. 1425. Both in China and Persia the knowledge of these properties may be much older. This knowledge was shared by some of the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history The history of Europe is traditionally divided into four time periods: prehistoric Europe (prior to about 800 BC), classical antiquity (800 BC to AD ...
mathematicians, and we see Pascal's triangle on the title page of Peter Apian's
German German(s) may refer to: * Germany (of or related to) ** Germania (historical use) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizens of Germany, see also German nationality law **Ge ...
arithmetic of 1527. After this, we find the triangle and the properties of binomial coefficients in several other authors."

Biographical film

In 2009,

IRIB The Islamic Republic of Iran Broadcasting (IRIB; fa, صدا و سيمای جمهوری اسلامی ايران, ''Sedā va Sīmā-ye Jomhūri-ye Eslāmi-ye Īrān'', , formerly called National Iranian Radio and Television until the Iranian rev ...

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'' (''Nardebām-e Āsmān'' ). 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.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, .

Notes

References

* * * * *

External links

*
PDF version
*
Mohammad K. Azarian, A summary of "Miftah al-Hisab", Missouri Journal of Mathematical Sciences, Vol. 12, No. 2, Spring 2000, pp. 75-95About Jamshid Kashani
* * * * {{DEFAULTSORT:Kashi, Jamshid 1380 births 1429 deaths People from Kashan 15th-century Iranian mathematicians Medieval Iranian astrologers 15th-century Iranian astronomers 15th-century astrologers Persian physicists Scholars from the Timurid Empire 15th-century inventors