I committed a mistake in my prior article regarding the history of calculus, mistakenly attributing the precision of Zu Chongzhi’s calculation of Pi. The Chinese mathematician Zu Chongzhi’s estimation of Pi was skillfully computed and remained significant for centuries. Nevertheless, it was not the most precise until Ludolph van Ceulen’s time in the sixteenth century. This was a major oversight due to unintentionally bypassing my source material.
Indeed, the remarkable contributions of Mādhava of Sangamagrāma, who lived from approximately 1340 to 1425 and established the Kerala School of astronomy and mathematics, came before van Ceulen’s work. Mādhava calculated Pi to eleven decimal places using an infinite series. A subsequent value is also credited to him with a precision of thirteen decimal places. An unnamed author from Kerala’s astronomical text, Karanapaddhati, further enhanced Pi, achieving accuracies of seventeen and even thirty-one decimal places through infinite series. This showcases a notable progress long before Europe recognized such precision in Pi calculations.
The contributions of the Kerala School were significant yet remained largely unrecognized outside the region until the nineteenth century. I intend to delve deeper into this school’s influential work in a forthcoming article.
Furthermore, the Persian astronomer and mathematician al-Kāshī (circa 1380–1429), a pivotal figure at Ulugh Beg’s observatory in Samarkand, greatly advanced the computation of Pi. In his “Treatise on the Circumference,” al-Kāshī skillfully applied Archimedes’ method of exhaustion, progressing to a polygon with 3 x 2^28 sides. He determined a value of 2π to nine sexagesimal digits, equivalent to sixteen decimal places (3.1415926535897932). Within the European framework—where the contributions of the Kerala School were unknown—al-Kāshī’s remarkable calculation maintained the record for the most accurate Pi value until van Ceulen.
These non-European contributions emphasize the rich and varied history of mathematical progress across different cultures long before they became widely recognized in Europe.