A Book Written at Twenty-Three
Aryabhata gives his own birth year, 476 CE, and states that he was twenty-three when he composed his work, which fixes it at 499. He says he wrote at Kusumapura, generally identified with Pataliputra, the old imperial capital near modern Patna, in the last decades of Gupta rule. Where he was actually from is disputed, with arguments made for Kerala and for the region around Ashmaka. This is more biography than survives for most figures of the period, and it is still almost nothing.
The Aryabhatiya is astonishingly compact: a hundred and twenty-one verses in four sections, covering a system of numerical notation, arithmetic and algebra, time reckoning, and the sphere. Verse was the medium of transmission because manuscripts were fragile and memory reliable, so the text is designed to be learned by heart. The consequence is extreme compression, and almost everything we understand of it depends on commentators, above all Bhaskara I, who explained it in 629 and preserved readings that would otherwise be lost.
Sines, Series and Approximation
The mathematical section is dense with results. He gives a table of half-chord values at intervals of three and three-quarter degrees, the foundation of the sine function; the Sanskrit term for it, jya, became Arabic jiba and then, by a misreading, Latin sinus, which is why we say sine at all. He gives correct formulas for the sums of consecutive integers, their squares and their cubes, methods for arithmetic progressions, quadratic problems, and the kuttaka procedure for solving indeterminate linear equations, which became a standard tool in Indian mathematics.
He states that a circle of diameter twenty thousand has a circumference of sixty-two thousand eight hundred and thirty-two, giving a value for pi of 3.1416, and he adds a word meaning approximate, which several scholars read as an awareness that the ratio cannot be given exactly. His system of place value with a decimal base is used throughout, though he encodes numbers alphabetically rather than with the numerals that later spread west. He also gave a rule for the volume of a sphere that is simply wrong, and later Indian mathematicians corrected it.
A Rotating Earth and Its Reception
His astronomy is where he broke hardest with tradition. He held that the apparent daily motion of the stars is caused by the rotation of the earth on its axis, using the image of a person in a moving boat seeing the riverbank slide backwards. He gave the length of the sidereal day to within a couple of seconds of the modern value, explained eclipses as shadows cast by the earth and the moon rather than as the demon Rahu of mythology, and stated that the planets and moon shine by reflected sunlight.
The rotation claim was rejected. Brahmagupta, a century later, attacked it directly and argued that a spinning earth would fling things off its surface, and the orthodox tradition followed him; some later commentators reinterpreted Aryabhata's verses to remove the doctrine. His work nonetheless became foundational, was translated into Arabic by around 800 as part of the transmission that carried Indian numerals and trigonometry to Baghdad and eventually to Europe, and India's first satellite, launched in 1975, was named for him.
