In a time and place where people believed certain distant stars, called "asuras," possessed malevolent powers capable of inflicting harm on Earth, Aryabhata the Elder took the first steps towards separating scientific explication from folklore and superstition. His Aryabhatiya was the first major book on Hindu mathematics, which summarized knowledge of his predecessors. While covering many aspects of arithmetic, algebra, and numerical notation, the majority of The Aryabhatiya dealt with trigonometric tables and formulae for use in astronomy. Any astronomical observations Aryabhata made were most likely completely unaided. Although The Aryabhatiya contained errors, it was translated and reproduced as Zij al-Arjabhar by the Arabs. One of Aryabhata's methods, a solution to the indeterminate quadratic xy = ax + by + c, was rediscovered by Leonhard Euler in the 18th century.
The City of Flowers
Aryabhata the Elder was born near what is now the city of Patna in India. His year of birth is sometimes cited as 475 but also possibly 476. Aryabhata's hometown was called Kusumapura, or The City of Flowers. The two major centers in the area represented the two intellectual threads he contended with in his lifetime. At that time, Patna was a royal seat, and according to legend it was founded by a knight with magic powers in honor of his princess, and blessed by the Buddha. Further away in Ujjian, the study of science and astronomy began to flourish, and this knowledge was being disseminated in the form of rhymed, romantic stories. Various mathematical problems, similar to those used in textbooks today, were solved in verse form used for social amusements. The public challenge and the romantic forms are combined in the most famous quotation from The Aryabhatiya, as Aryabhata the Elder commands a "beautiful maiden" to answer a problem that requires inversion.
The Aryabhatiya was produced in the year 499. It described the Indian numerical system with nine symbols, and listed various rules for arithmetic and trigonometric calculations. It also made use of continued fractions, square and cube roots, and the sine function when needed. Solutions were given for linear and quadratic equations and diophantine equations of the first degree. These involved one of the first recorded uses of algebra and decimal place-value. Unlike the Greeks, the Hindus solved diophantines for all possible integral solutions, as they were more tolerant of negative, irrational, and other such numbers. For instance, one value for π given by Aryabhata is the square root of 10, generally called "the Hindu value."
Pebbles and Crystals
The combination of correct and incorrect answers to the major questions of its time led one Arabic commentator, al-Biruni, to describe Hindu mathematics as a mixture of "common pebbles and costly crystals." Aryabhata gave an accurate approximation of π, although he overestimated the length of the year by 12 minutes and 30 seconds. He was singular in describing the orbits of the planets as ellipses. In the Ganita, a poem composed in 33 couplets, he correctly states the formulae for areas of a triangle and a circle. However, when Aryabhata attempts to extrapolate those to figure the volumes of three-dimensional shapes in the same couplets, he is not successful. Nonetheless, Aryabhata the Elder's commitment to general methods caused him to apply what is now nicknamed "the pulverizer." This rule finds the greatest common divisor of a and b by division, equivalent to Euler's later version of reducing a over b to a continued fraction. While "the pulverizer" has also been known as the Diophantine method, the fact that Diophantus himself never used it renders the term a misnomer.
Aryabhata died around 550, though it is not known where. As an astronomer he argued--against Vedic tradition--that the Earth was round and rotated daily. He correctly explained why equinoxes, solstices, and eclipses occur. These ideas were not accepted in Aryabhata's lifetime, but his mathematics had set the foundation for developments in the Eastern and Western worlds for centuries to come. In India particularly, Aryabhata the Elder marked the end of the sacred or "S'ulvasutra" period, during which mathematics was used primarily by priests for temple architecture. He ushered in the "astronomical period" that lasted until the year 1200.
The Aryabhatiya held the same stature in India that Euclid's Elements did in ancient Greece. Bhaskara I wrote a commentary on this work in 629. Aryabhata also influenced the work of Hindu astronomer Brahmagupta.
- The Aryabhatiya of Aryabhata. Edited by W.E. Clark, 1930.
- Boyer, Carl B. A History of Mathematics. New York: John Wiley & Sons, 1968, pp. 229-241.
- Cajori, Florian. "The Hindus." A History of Mathematics. Second Edition, revised and enlarged. New York: Macmillan, 1919, pp. 83-98.
- Eaves, Howard. Great Moments in Mathematics (Before 1650). Mathematical Association of America, Inc., 1980, pp. 21-2, 105.
- Eaves, Howard. An Introduction to the History of Mathematics. Fourth Edition. Holt, Rinehart & Winston, 1976, pp. 180-87.
- Wu, Frank. "Aryabhata." The Great Scientists. Edited by Frank N. Magill. Danbury, CT: Grolier, 1989, pp. 137-141.
- Datta, B. "Two Aryabhatas of al-Biruni." Bulletin of Calcutta Mathematical Society17 (1926): 59-74.
- O'Connor, John J. and Robertson, Edmund F. "Aryabhata the Elder." MacTutor History of Mathematics Archive. http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Aryabhata.html (July 1997).