Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī) (Persian: غیاثالدین جمشید کاشانی Ghiyās-ud-dīn Jamshīd Kāshānī; c. 1380 – 22 June 1429) was a Persian astronomer and 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 Much of al-Kāshī's work was not brought to Europe and still, even the extant work, remains unpublished in any form.
Biography
Al-Kashi was born in 1380, in Kashan, in central Iran, to a Persian family. 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, in 1429. Some state that he was murdered and say that Ulugh Beg probably ordered this, whereas others suggest he died a natural death. Regardless, after his death, Ulugh Beg described him as "a remarkable scientist" who "could solve the most difficult problems".
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.
Sizes and distances of heavenly bodies He wrote the book Sullam al-sama' (Stairway to the Heavens, 1407) on the resolution of difficulties met by predecessors in the determination of sizes and distances of heavenly bodies such as the Earth, the Moon, the Sun, and the stars.
Astronomical 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
Computation of 2π Al-Kashi made the most accurate approximation of π to date in his al-Risāla al-muhītīyya (Treatise on the Circumference). 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, 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 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 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 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-Din al-Tusi. Al-Kāshī improved on this by using a form of Newton's method to solve x P − N = 0 {\displaystyle 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. In order to determine sin 1°, al-Kashi discovered the following formula, often attributed to François Viète in the sixteenth century:
sin 3 ϕ = 3 sin ϕ − 4 sin 3 ϕ {\displaystyle \sin 3\phi =3\sin \phi -4\sin ^{3}\phi \,\!}
The Key to Arithmetic
Law of cosines
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