Aryabhata Calculates Pi

In 499 AD, the Indian mathematician and astronomer Aryabhata completed the Aryabhatiya, a compact Sanskrit work that gathered the mathematics and astronomy of his day. It gave an unusually accurate approximation of pi and argued that the apparent daily motion of the stars is caused by the Earth turning on its axis. Together, these ideas make the book one of the landmark scientific texts of the ancient world.

Aryabhata was born in 476 AD, and later tradition places him at Kusumapura, usually identified with the region around Pataliputra (modern Patna), then a major center of learning under the Gupta Empire. He wrote at a time when Indian scholars were absorbing and refining earlier astronomical traditions, including influences that had reached India from the Greek-speaking world. The Aryabhatiya is brief, written in about 120 verses of dense, memorizable Sanskrit, and was probably meant to be studied alongside a teacher's commentary.

Its rule for the circle is the best-known result. Aryabhata states that if a circle has a diameter of 20,000, its circumference is approximately 62,832. That gives a value for pi of about 3.1416, correct to four decimal places. He used the word "asanna," meaning "approaching," which some later commentators have read as a sign that he knew the value was only an approximation. This reading is debated, and the evidence is not conclusive. How he arrived at the number is also unclear, since the text gives no derivation. The book also contains work on square and cube roots, arithmetic and geometric progressions, a table of sines, and methods for solving certain indeterminate equations.

In astronomy, Aryabhata held that the Earth rotates daily, comparing the apparent motion of the stars to how objects on the bank seem to move backward to someone traveling in a boat. This was a bold departure from the common view of a stationary Earth. He also explained that the Moon and planets shine by reflected sunlight, and he offered a largely correct account of solar and lunar eclipses as shadow effects rather than the work of mythic demons. His planetary model was still Earth-centered, though, and it should not be mistaken for the later heliocentric theory.

The Aryabhatiya was widely studied and commented on across India, and its methods shaped later astronomers such as Bhaskara I and Brahmagupta, although some of them disagreed with his rotating Earth. Its ideas also traveled through Arabic translations into the Islamic world. Today Aryabhata is honored as a central figure in the history of mathematics, and India named its first national satellite, launched in 1975, after him.