Aryabhata writes the Aryabhatiya

The work is short: a 13-verse introduction, which sets out the astronomical constants and a table of sines in a compact alphabetic notation for numbers, followed by 108 verses on mathematics (ganita), the reckoning of time (kalakriya) and the sphere (gola). Aryabhata gives rules for square and cube roots, areas, arithmetic series and the sums of squares and cubes, states the value of π as 62,832/20,000 = 3.1416, calling it ‘approximate’, and describes the kuttaka (‘pulverizer’) method for indeterminate equations. His astronomy is geocentric, with epicycles, but he taught that the apparent daily rotation of the heavens is caused by the Earth turning on its axis — as a man in a boat moving forward sees stationary objects on the shore moving backward — explained eclipses by the shadows of the Earth and the Moon, held that the Moon and the planets are lit by the Sun, and gave the sidereal year as 365 days 6 hours 12 minutes 30 seconds.

The date comes from a verse of the third chapter in which Aryabhata says that twenty-three years of his life had passed when 3,600 years of the Kali age had elapsed — that is, in 499. This is usually taken as the year of writing, but the verse gives his age rather than the date of the work, and the historian of astronomy Roger Billard, who showed that Aryabhata’s planetary parameters rest on observations made in his own time, placed the treatise around 510. The Aryabhatiya was criticised by Brahmagupta in 628 and explained by commentators from Bhaskara I (629) to the Kerala astronomers of the 15th century, and Arabic astronomers knew its methods. The word ‘sine’ goes back to its ardha-jya, ‘half-chord’: the Arabic borrowing jiba was read as jaib, ‘fold’, and translated into Latin as sinus.