Aryabhatiya
Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature, and has survived to modern times.
The text is written in Sanskrit and structured into four sections, overall covering 121 verses that describe different results using a mnemonic style typical of the Indian tradition. 33 verses are concerned with mathematical rules.
The four chapters elaborate on the following details:
(i) the astronomical constants and the sine table
(ii) mathematics required for computations
(iii) division of time and rules for computing the longitudes of planets using eccentrics and ellipses
(iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses
The treatise uses a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Pait?mahasiddh?nta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger ??ghra (fast) epicycle.
It has also been interpreted as advocating Heliocentrism, where Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun (according to this view, it was heliocentric). Aryabhata asserted that the Moon and planets shine by reflected sunlight and that the orbits of the planets are ellipses. He also correctly explained the causes of eclipses of the Sun and the Moon. His value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the true value of 365 days 6 hours 9 minutes 10 seconds. In this book, the day was reckoned from one sunrise to the next, whereas in his "?ryabhata-siddh?nta" he took the day from one midnight to another. There was also difference in some astronomical parameters.
A close approximation to pi is given as : "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, ? ?62832/20000 = 3.1416, correct to four rounded-off decimal places.
Aryabhata was the first astronomer to make an attempt at measuring the Earth's circumference since Erastosthenes (circa 00 BC). Aryabhata accurately calculated the Earth's circumference as 24,835 miles, which was only 0.2%smaller than the actual value of 24,902 miles. This approximation remained the most accurate for over a thousand years.
Aryabhata's methods of astronomical calculations have been in continuous use for practical purposes of fixing the Panchanga (Hindu calendar).