Al-Kashi

Al-Kashi (1380 - 1429) was a renowned mathematician and astronomer in early fifteenth-century Persia and Central Asia. An Iranian from a humble background, he was entirely self-taught, and was one of the leading scholars at the newly created University of Samarkand in what is present-day Uzbekistan. In 1424 Al-Kashi published a treatise on circumference, in which he calculated "pi", the ratio of a circle's circumference to its diameter, to nine decimal places. Nearly two hundred years would pass before another mathematician surpassed this achievement.

There are various spellings given for Al-Kashi's full name, but the standard English transliteration seems to be Ghiyath al-Din Jamshid Mas'ud al-Kashi. He was born in 1380 in Kashan, a desert town in Iran located near the Central Iranian Range. Kashan is a noted oasis on the road to Qom, the Shiite holy city in Iran, and archaeological discoveries cite Al-Kashi's birthplace as one the oldest inhabited places on Earth. During his childhood, however, Kashan and the surrounding area were subject to periodic raids by the conqueror Tamerlane, an Uzbek of Mongol heritage and a Muslim. Some ten years before Al-Kashi's birth, Tamerlane founded his empire, which was a restoration of a previous Mongol kingdom, at the city of Samarkand, one of the oldest inhabited urban centers in the world.

With Peace, Economic Prosperity

Tamerlane began conquering territory to the west and south, and his move into Persia in 1383 began a period of difficulty for families like Al-Kashi's. The people lived in poverty for a number of years, and were forced to move frequently due to military raids. During this time Al-Kashi taught himself mathematics and astronomy, though the possible written sources he may have used are unknown. When the emperor Tamerlane died in 1405, his fourth son, Shah Rokh, ascended to the throne of the eastern portion of the Timurid empire, which encompassed Persia and Transoxania. This ushered in a more stable period, and one in which the economic climate vastly improved. Shah Rokh and his wife, a Persian princess named Gauhar Shad, were also enthusiastic supporters of the arts and sciences, and this period became a time of intellectual accomplishment and fervor in the region that gave scholars like Al-Kashi fertile soil in which to flourish.

By the time Al-Kashi reached adulthood, the Arabic world had produced a number of great mathematicians over the past millennia. The ancient Greeks formulated many of the algebraic and geometry theories still in use in modern times, but further scholarship died out with the rise of Christianity in Europe and the Mediterranean area during the early medieval period - mathematics and astronomy were closely tied to one another, and studying the heavens was viewed as the devil's work. But Islamic centers of learning flourished during this period, and several notable men made important discoveries in mathematics in Cairo, Baghdad, and the cities of Moorish Spain.

In 1406, the year Al-Kashi turned 26, he wrote about a lunar eclipse he had observed. By this time he was already writing his first book, which he finished in Kashan in March of 1407. It was titled Sullam al-sama (The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes). In 1409 Shah Rokh's oldest son, Ulugh Beg (1393 - 1449), became ruler of the Transoxania portion of the Timurid empire. Ulugh Beg was a noted mathematician and astronomer, and at Samarkand he began to lay the foundations for what would become this part of the world's most esteemed university.

Courted Ulugh Beg's Favor

When Al-Kashi finished his Mukhtasar dar 'ilm-i hay'at (Compendium of the Science of Astronomy) in 1411, he dedicated it to Iskander, a Timurid ruler in Iran. This was a common practice, because scholars relied on royal patronage in order to carry out their work and earn a living through it. Iskander was murdered in 1414, the same year that Al-Kashi completed Khaqani Zij. This title reflected a new patron in 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. These were used to calculate the coordinates in the heavens, helped astronomers measure distances, and predicted the motion of the sun, moon, and planets, as well as longitudinal and latitudinal parallaxes. The Islamic world was profoundly interested in such navigational aids because of the qibla, the direction that a Muslim needed to face for prayer. Since 624 CE, devout Muslims who followed the five pillars of their religion had been instructed to face the Saudi Arabian city of Mecca, Islam's holy city, five times daily when they prayed. Many Muslims were traders, or traveled on other business, and used a complex measuring device called an astrolabe to find the direction of Mecca so that they could fulfill their religious obligation without error.

Al-Kashi did not yet have a formal patron, but it is known that he spent time in Herat, an ancient city in western Afghanistan that dated back to 500 BCE and was a renowned center for the production of bronze artifacts. In 1416 he 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 Garden Excursion, on the Method of Construction of the Instrument Called Plate of Heavens). The latter work contains a description of his invention for a device to predict the positions of the planets.

Ulugh Beg invited Al-Kashi to teach at the University of Samarkand. He became its leading astronomer, and later in the century was described by a historian as the second Ptolemy, referring to the second-century Greek astronomer who lived and worked in Alexandria, Egypt, when it was the greatest center of scientific scholarship. Ptolemy preserved what had been known about the stars since the first Greek astronomers, named some 48 constellations in the night sky, and devised navigational tables that were used by mariners well into the 1600s.

Rose to Prominence

Al-Kashi wrote about his life in Samarkand in letters back to his father, and these provide contemporary scholars an unusual glimpse into this time and place. He wrote of the observatory that Ulugh Beg had built at Samarkand in 1424. Known as Gurkhani Zij, it featured an immense astrolabe with a precision-cut arc made of marble that was 63 yards long.

That same year 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 nine decimal places. The last reliable pi figure dated from nearly 900 years earlier, and had been ascertained by Chinese astronomers of the fifth century, but it was only to six decimal places. It would be nearly two hundred years before another mathematician found a more accurate calculation for pi, and that was the German-born mathematician Ludolph van Ceulen (1540 - 1610), who calculated it to 20 decimal places. Van Ceulen lived in the Dutch cities of Delft and Leiden for many years, and his most famous work, Van den Circkel (On the Circle), was published in 1596.

In 1427 Al-Kashi wrote another important text, Miftah al-Hisab (The Key to Arithmetic). This was intended to serve as a textbook for scholars at Samarkand, providing basic and advanced math for astronomy, but it was also designed for use by students of architecture, land surveying, accounting, and commerce. It was notable for its inclusion of decimal fractions. These had been worked out a few centuries earlier by mathematicians of the school of al-Karaji (Abu Bakr ibn Muhammad ibn al-Husayn Al-Karaji, 953-1029).

Calculated Muqarna Surface

One of the more impressive sections of The Key to Arithmetic 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, for craftspeople were not paid by the hour in this era. "Payment per cubit was common in Ottoman architectural practice," noted a University of Heidelberg scholar, Yvonne Dold-Samplonius, in the Nexus Network Journal, "where a team of architects and surveyors had to make cost estimates of projected buildings and supply preliminary drawings for various options. In addition to facilitating estimates of wages and building materials before construction, Al-Kashi's formulas may also have been used in appraising the price of a building after its completion."

Al-Kashi's last work was Risala al-watar wa'l-jaib (The Treatise on the Chord and Sine), but it was unfinished at the time of his death in 1429. It was completed by Qadi Zada al-Rumi, another renowned mathematician at Samarkand, and includes sine calculations and a discussion of cubic equations. After Al-Kashi's death, Ulugh Beg praised him as "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," according to a biography that appeared on the website of the School of Mathematics and Statistics of the University of St. Andrews in Scotland. Contemporary scholars, however, believe that Al-Kashi was murdered on orders from Ulugh Beg.

The buildings of Ulugh Beg's university in Samarkand survive as part of the Registan, the old commercial center of the city. The observatory, Gurkhani Zij, was lost for a number of centuries, but its ruins were unearthed in 1908.

Books

Science and Its Times, Vol. 2: 700-1450, Gale Group, 2001.

Online

"Calculation of Arches and Domes in 15th Century Samarkand," Nexus Network Journal, http://www.nexusjournal.com/conferences/N2000-DoldSamplonius.html (January 12, 2006).

"Ghiyath al-Din Jamshid Mas'ud al-Kashi," School of Mathematics and Statistics, University of St. Andrews in Scotland, http://www-groups.dcs.st-and.ac.uk/∼history/Mathematicians/Al-Kashi.html (January 2, 2006).