# Ghiyath al-Din Jamshid Mas’ud al-Kashi

### Born: about 1380 in Kashan, Iran Died: 22 June 1429 in Samarkand, Transoxania (now Uzbek)

Details of Jamshid al-Kashi‘s life and works are better known than many others from this period although details of his life are sketchy. One of the reasons we is that he dated many of his works with the exact date on which they were completed, another reason is that a number of letters which he wrote to his father have survived and give fascinating information.

Al-Kashi was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range. At the time that al-Kashi was growing up Timur (often known as Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh.

While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town. Conditions improved markedly when Shah Rokh took over after his father’s death. He brought economic prosperity to the region and strongly supported artistic and intellectual life. With the changing atmosphere, al-Kashi’s life also improved markedly. The first event in al-Kashi’s life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406.

It is reasonable to assume that al-Kashi remained in Kashan where he worked on astronomical texts. He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived. The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies). At this time it was necessary for scientists to obtain patronage from their kings, princes or rulers. Al-Kashi played this card to his advantage and brought himself into favour in the new era where patronage of the arts and sciences became popular. His Compendium of the Science of Astronomy written during 1410-11 was dedicated to one of the descendants of the ruling Timurid dynasty.

Samarkand, in Uzbekistan, is one of the oldest cities of Central Asia. The city became the capital of Timur’s empire and Shah Rokh made his own son, Ulugh Beg, ruler of the city. Ulugh Beg, himself a great scientist, began to build the city into a great cultural centre. It was to Ulugh Beg that Al-Kashi dedicated his important book of astronomical tables Khaqani Zij which was based on the tables of Nasir al-Tusi. In the introduction al-Kashi says that without the support of Ulugh Beg he could not have been able to complete it. In this work there are trigonometric tables giving values of the sine function to four sexagesimal digits for each degree of argument with differences to be added for each minute. There are also tables which give transformations between different coordinate systems on the celestial sphere, in particular allowing ecliptic coordinates to be transformed into equatorial coordinates. See [14] for a detailed discussion of this work.

The Khaqani Zij also contains [1]:-

… detailed tables of the longitudinal motion of the sun, the moon, and the planets. Al-Kashi also gives the tables of the longitudinal and latitudinal parallaxes for certain geographical latitudes, tables of eclipses, and tables of the visibility of the moon.

Al-Kashi had certainly found the right patron in Ulugh Beg since he founded a university for the study of theology and science at Samarkand in about 1420 and he sought out the best scientists to help with his project. Ulugh Beg invited Al-Kashi to join him at this school of learning in Samarkand, as well as around sixty other scientists including Qadi Zada. There is little doubt that al-Kashi was the leading astronomer and mathematician at Samarkand and he was called the second Ptolemy by an historian writing later in the same century.

Letters which al-Kashi wrote in Persian to his father, who lived in Kashan, have survived. These were written from Samarkand and give a wonderful description of the scientific life there. In 1424 Ulugh Beg began the construction of an observatory in Samarkand and, although the letters by al-Kashi are undated they were written at a time when construction of the observatory had begun. The contents of one of these letters has only recently been published, see [8].

In the letters al-Kashi praises the mathematical abilities of Ulugh Beg but of the other scientists in Samarkand, only Qadi Zada earned his respect. Ulugh Beg led scientific meetings where problems in astronomy were freely discussed. Usually these problems were too difficult for all except al-Kashi and Qadi Zada and on a couple of occasions only al-Kashi succeeded. It is clear that al-Kashi was the best scientist and closest collaborator of Ulugh Beg at Samarkand and, despite al-Kashi’s ignorance of the correct court behaviour and lack of polished manners, he was highly respected by Ulugh Beg. After Al-Kashi’s death, Ulugh Beg described him as (see for example [1]):-

… a remarkable scientist, one of the most famous in the world, who had a perfect command of the science of the ancients, who contributed to its development, and who could solve the most difficult problems.

Although al-Kashi had done some fine work before joining Ulugh Beg at Samarkand, his best work was done while in that city. He produced his Treatise on the Circumference in July 1424, a work in which he calculated 2π to nine sexagesimal places and translated this into sixteen decimal places. This was an achievement far beyond anything which had been obtained before, either by the ancient Greeks or by the Chinese (who achieved 6 decimal places in the 5th century). It would be almost 200 years before van Ceulen surpassed Al-Kashi’s accuracy with 20 decimal places.

Al-Kashi’s most impressive mathematical work was, however, The Key to Arithmetic which he completed on 2 March 1427. The work is a major text intended to be used in teaching students in Samarkand, in particular al-Kashi tries to give the necessary mathematics for those studying astronomy, surveying, architecture, accounting and trading. The authors of [1] describe the work as follows:-

In the richness of its contents and in the application of arithmetical and algebraic methods to the solution of various problems, including several geometric ones, and in the clarity and elegance of exposition, this voluminous textbook is one of the best in the whole of medieval literature; it attests to both the author’s erudition and his pedagogical ability.

Dold-Samplonius has discussed several aspects of al-Kashi’s Key to Arithmetic in [11], [12], and [13]. (see also [3]). For example the measurement of the muqarnas refers to a type of decoration used to hide the edges and joints in buildings such as mosques and palaces. The decoration resembles a stalactite and consists of three-dimensional polygons, some with plane surfaces, and some with curved surfaces. Al-Kashi uses decimal fractions in calculating the total surface area of types of muqarnas. The qubba is the dome of a funerary monument for a famous person. Al-Kashi finds good methods to approximate the surface area and the volume of the shell forming the dome of the qubba.

We mentioned above al-Kashi’s use of decimal fractions and it is through his use of these that he has attained considerable fame. The generally held view that Stevin had been the first to introduce decimal fractions was shown to be false in 1948 when P Luckey (see [4]) showed that in the Key to Arithmetic al-Kashi gives as clear a description of decimal fractions as Stevindoes. However, to claim that al-Kashi is the inventor of decimal fractions, as was done by many mathematicians following the work of Luckey, would be far from the truth since the idea had been present in the work of several mathematicians of al-Karaji’s school, in particular al-Samawal.

Rashed (see [5] or [6]) puts al-Kashi’s important contribution into perspective. He shows that the main advances brought in by al-Kashi are:-

(1) The analogy between both systems of fractions; the sexagesimal and the decimal systems.
(2) The usage of decimal fractions no longer for approaching algebraic real numbers, but for real numbers such as π.

Rashed also writes (see [5] or [6]):-

… Al-Kashi can no longer be considered as the inventor of decimal fractions; it remains nonetheless, that in his exposition the mathematician, far from being a simple compiler, went one step beyond al-Samawal and represents an important dimension in the history of decimal fractions.

There are other major results in the work of al-Kashi which were pointed out by Luckey. He found that al-Kashi had an algorithm for calculating nth roots which was a special case of the methods given many centuries later by Ruffini and Horner. In later work Rashed shows (see for example [5] or [6]) that Al-Kashi was again describing methods which were present in the work of mathematicians of al-Karaji’s school, in particular al-Samawal.

The last work by al-Kashi was The Treatise on the Chord and Sine which may have been unfinished at the time of his death and then completed by Qadi Zada. In this work al-Kashi computed sin 1° to the same accuracy as he had computed π in his earlier work. He also considered the equation associated with the problem of trisecting an angle, namely a cubic equation. He was not the first to look at approximate solutions to this equation since al-Biruni had worked on it earlier. However, the iterative method proposed by al-Kashi was [1]:-

… one of the best achievements in medieval algebra. … But all these discoveries of al-Kashi’s were long unknown in Europe and were studied only in the nineteenth and twentieth centuries by … historians of science….

Let us end with one final comment on the al-Kashi’s work in astronomy. We mentioned earlier the astronomical tables Khaqani Zij produced by al-Kashi. It is worth noting that Ulugh Begalso produced astronomical tables and sine tables, and it is almost certain that these tables were based on al-Kashi’s tables and almost certainly produced with al-Kashi’s help.

Ghiyāth al-Dīn Jamshīd Masʿūd al-Kāshī (or al-Kāshānī)[1] (Persian: غیاث‌الدین جمشید کاشانی‎ Ghiyās-ud-dīn Jamshīd Kāshānī) (c. 1380 Kashan, Iran – 22 June 1429 Samarkand, Transoxania) was a Persianastronomer and mathematician.

## Biography

Al-Kashi was one of the best mathematicians in the Islamic world. He was born in 1380, in Kashan, in central Iran. This region was controlled by Tamurlane, better known as Timur. 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. 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 Samarkandwhich 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 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.[2]

### 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–altitudeinstrument he invented, and a small armillary sphere incorporating an alhidade which he invented.[3]

#### 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,[4] and for performing linear interpolation.[5]

#### 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,[5] and the planets in terms of elliptical orbits;[6] the latitudes of the Sun, Moon, and planets; and the ecliptic of the Sun. The instrument also incorporated an alhidade and ruler.[7]

## 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.[2]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.[8]In order to determine sin 1°, al-Kashi discovered the following formula often attributed to François Viète in the 16th century:[9]
$\sin 3 \phi = 3 \sin \phi - 4 \sin^3 \phi\,\!$

### The Key to Arithmetic

#### Computation of 2π

In his numerical approximation, he correctly computed 2π (or $\tau$) to 9 sexagesimal digits[10] in 1424,[2]and he converted this approximation of 2π to 17 decimal places of accuracy.[11] 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.[2] It should be noted that al-Kashi’s goal was not to compute the circle constant with as many digits as possible but to compute it 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):[12]
“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).[13]

#### Khayyam’s triangle

In considering Pascal’s triangle, known in Persia as “Khayyam’s triangle” (named after Omar Khayyám), Struik notes that (p. 21):[12]
“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.[14]The properties of binomial coefficients were discussed by the Persian mathematician Jamshid Al-Kāshī in his Key to arithmetic of c. 1425.[15] 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.[16]

## 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 [17][18] (Nardebām-e Āsmān[19]). 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.[20][21][22]