Jamshid al-Kashi

Key Accomplishments

• Calculated pi to 16 decimal places, a record of precision unmatched for nearly 180 years
• Computed the sine of one degree to 18 decimal places, the most accurate trigonometric value of his era
• Authored Miftah al-Hisab (Key of Arithmetic), a comprehensive mathematical textbook that standardized decimal fractions
• Served as the leading mathematician at Ulugh Beg's observatory in Samarkand

The Full Story

Al-Kashi was born in Kashan, Iran, and spent his early career in poverty, struggling to find patronage during the turbulent post-Timurid period. His fortunes changed when Ulugh Beg, the scholarly Timurid prince and governor of Samarkand, invited him to join the new observatory he was building. Al-Kashi became the leading mathematician at the observatory and its effective scientific director.

His most famous achievement was calculating the value of pi to 16 decimal places using a polygon with 3 x 2^28 sides, presented in his Treatise on the Circumference (Risala al-Muhitiyya) in 1424. This was far more precise than any previous calculation and the record was not broken until the French mathematician Viete in 1579. He also computed the sine of one degree to 18 decimal places, providing the most accurate trigonometric table of the medieval period.

Al-Kashi's Miftah al-Hisab (Key of Arithmetic), completed in 1427, was a comprehensive textbook covering arithmetic, algebra, and mensuration that became a standard reference across the Islamic world. In it, he systematized the use of decimal fractions, showing how they could simplify complex calculations. He also developed an iterative method for solving cubic equations that anticipated later European techniques. His letter to his father describing life at the Samarkand observatory provides a rare and valuable glimpse into the daily work of medieval Islamic scientists.

Places

Connected Figures

Sources

• [1]Katz, Victor J. A History of Mathematics: An Introduction. Addison-Wesley, 2008.
• [2]Kennedy, E.S. "The Exact Sciences in Timurid Iran." Cambridge History of Iran, Vol. 6, 1986.
• [3]al-Kashi, Jamshid. Miftah al-Hisab (Key of Arithmetic), trans. Nader Nabulsi. Springer, 2019.