Muslim Scientists and Thinkers–Jamshid al-Kashi

By Syed Aslam, science@muslimobserver.com

Jamshid al-Kashi was born in Kashan, which lies in the eastern foot of the Central Iranian Range in 1380 CE.  When al-Kashi was growing up, Timur Leng was the ruler of that area, and he was not interested in science and education. Timur died in 1405 and his empire was divided between his two sons, one of whom was    Shah Rukh.

During Timur’s reign, conditions were difficult, with widespread poverty. Al-Kashi lived in poverty, but devoted himself to astronomy and mathematics while moving from town to town. Conditions improved when Shah Rukh became the ruler. He brought economic prosperity to the region, and strongly supported science and intellectual activity. With the change in economic conditions, al-Kashi’s life also improved markedly.

Samarkand Uzbekistan became the capital of the empire, and Shah Rukh made his son, Ulugh Beg, ruler of the city. Ulugh Beg, himself a great scientist, began to build the city into a great center of learning.  He invited al-Kashi  to Samarkand to work with him which he gladly accepted. He lived there and worked on many books till he died in 1429 CE.

Al Kashi was a great astronomer and mathematician who made original contributions in both field.  He wrote his first book on astronomy when he was very young, titled Sullam al-sama (The Stairway of Heaven) In this book he discussed how to determine altitude, size and distance of the heavenly bodies.  He   finished his second book on astronomy;  Mukhtasar dar ‘ilm-i hay’at (Compendium of the Science of Astronomy) and  dedicated it to  a Timurid ruler of Iran.  

Al-Kashi’s third book,  Khaqani Zij, was completed at Samarkand and dedicated to  his new patron Ulugh Beg, who was also known as Khaqani, or “Supreme Ruler”; zij was the Persian term for astronomical tables. Al-Kashi’s astronomical tables were based on an earlier work done by another Persian, Nasir al-Tusi. This book had a useful tool to calculate the coordinates  of the heavens, and helped astronomers measure distances, and predicted the motion of the sun, moon, and planets, as well as longitudinal and latitudinal parallaxes.

In 1416 CE al-Kashi completed two new works, Risala dar sharh-i alat-i rasd (Treatise on the Explanation of Observational Instruments) and Nuzha al-hadaiq fi kayfiyya san’a al-ala almusamma bi tabaq al-manatiq (The  Method of Construction of the Instrument Called Plate of Heavens). This book contains a description of his invention, a device to predict the positions of the planets.

Although al-Kashi had done fine work in mathematics before joining Ulugh Beg at Samarkand, his best work was done while he was in that city. In 1424 CE al-Kashi finished his most famous work, the Risala al-muhitiyya (Treatise on the Circumference). In it he calculated pi, (the ratio of a circle’s circumference to its diameter) to sixteen decimal places. The last reliable pi figure  had been calculated by Chinese astronomers in the fifth century, but it was only good to six decimal places. It would be nearly two hundred years before European mathematicians found a more accurate calculation for pi,  which was accurate to 20 decimal places.

Al-Kashi’s most impressive mathematical work was the book he wrote titled;  The Key to Arithmetic, intended for those studying astronomy, accounting, trading, students of architecture, and land surveying. It was notable for its inclusion of decimal fractions. One of the most impressive sections of this book was al-Kashi’s formula for measuring a complex shape called a muqarna. The muqarna was a standard form used by Arabic world architects to hide edges and joints in mosques, palaces, and other large public buildings. It was a three-dimensional polygon or wedge form combined into honeycomb patterns. Al-Kashi’s muqarna measurement had a practical application in the field of civil and architectural engineering.  

Al-Kashi’s important and last work in the field of mathematics was his book titled;  The Treatise on the Chord and Sine, in this work al-Kashi computed sin 1° to the same accuracy as he had computed pi in his earlier work. In order to determine sin1 value he discovered a new trigonometric formula  sin 3 φ = 3 sin φ – 4 sin3 φ.     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-Birunid worked on it earlier. However, the iterative method proposed by al-Kashi was one of the best achievements in medieval algebra.

Al-Kashi was the first mathematician to discover the law of cosines in a form suitable of triangulation which was later called; the ‘Theorem of Al-Kashi’.

Al-Kashi wrote many letters to his father about his life in Samarkand, and some have survived–this provides an unusual glimpse into his contemporary scholars and scientists. He wrote about the observatory that Ulugh Beg had built at Samarkand in 1424 CE,  it featured an immense astrolabe with a precision-cut marble arc, 62 yards long.

Al-Kashi was a brilliant astronomer and mathematician.  After al-Kashi’s death, Ulugh Beg, the ruler, praised him as a remarkable scientist, one of the most famous in the area, who had a perfect command of the science of the ancients, who made original  contributions in the field astronomy and mathematics, and who could solve the most difficult problems.

Aslamsyed1@yahoo.com