Born : c. 1380, Kashan, Iran
Died : 22 June 1429 (aged 48)
Main interest(s) : Astronomy, Mathematics.
Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (Persian: غیاث
الدین جمشید کاشانی Ghiyās-ud-dīn
Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania)
was a Persian astronomer and mathematician.
Much of al-Kāshī's work was not
brought to Europe, and much, even the extant work, remains unpublished in any
form.
Biography
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, 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, 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 Ulugh Beg, who invited al-Kashi to work at his
observatory (see Islamic astronomy) and his university (see Madrasah) which
taught theology. 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.
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, 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 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.
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.
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, and the planets in terms of
elliptical orbits; the latitudes of the Sun, Moon, and planets; and the
ecliptic of the Sun. The instrument also incorporated an alhidade and ruler.
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 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 π, 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 and numerical analysis, he developed an iterative
method for solving cubic equations, 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. Al-Kāshī improved on this by using a form of Newton's method to solve to
find roots of N. In western Europe, a similar method was later
described by Henry 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 in the sixteenth century:
The Key
to Arithmetic
Computation
of 2π
In his numerical approximation, he correctly computed 2π to 9 sexagesimal digits in
1424, and he converted this estimate of 2π to 16 decimal places
of accuracy. This was far more accurate than the estimates earlier given
in Greek mathematics (3 decimal places by Ptolemy, 150
CE), Chinese mathematics (7 decimal places by Zu Chongzhi, 480
Ad) or Indian mathematics (11 decimal places by Madhava of
Sangamagrama, c. 1400). The accuracy of al-Kashi's estimate
was not surpassed until Ludolph van Ceulen computed 20 decimal places
of π 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 highest desirable precision (the
diameter of a hair).
Decimal
fractions
In discussing decimal fractions, Struik states that
(p. 7):
"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 his Key to arithmetic (Samarkand, early fifteenth
century)."
Khayyam's
triangle
In considering Pascal's triangle, known in Persia as "Khayyam's
triangle" (named after Omar Khayyám), Struik notes that :
"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. 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 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.
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
(Nardebām-e Āsmān). 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.
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