Aryabhata (Sanskrit: आर्यभट, 476 - 550) was the first of the major mathematical astronomers, and Indian astronomers. His work includes Āryabhaṭīya (499, when he was 23 years old) and Arya-siddhanta.
In his book, "Aryabhatiya", mathematical and astronomical theories presented the Earth rotating in its axis and the periods of the planets were given with respect to the sun (in other words, it was heliocentric). He believed that the Moon and planets shine due to light reflected solar and he believed that the orbits of the planets would be elliptical. The book explains the causes of eclipses the Sun and the Moon correctly. Its value for the duration of the year in 365 days, 6 hours, 12 minutes and 30 seconds is remarkably close to the true value which is approximately 365 days and 6 hours. This book is divided into four chapters: (i) astronomical constants and sine table (ii) mathematics used in computing (iii) division of time and rules for calculating planet lengths using eccentrics and epicycles (iv) the armillary sphere, rules related to trigonometry problems and the computation of eclipses. In this book, the day was considered from one dawn to the next, while in his "Arya-siddhanta" the day was taken from one midnight to the next. There is also difference in some astronomical parameters.
He was the first to explain how lunar and solar eclipses occur.
Aryabhata also gave a very close indication to Pi. In the Ariaria he recorded: "Add four to a 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 ratio of the circumference to the diameter is given. In other words, π ≈ 62832/20000 = 3.1416, correct for the four decimal places.
Aryabhata was the first astronomer to attempt to measure the circumference of the Earth since Eratosthenes (circa 200 BC), calculating the circumference of the planet at 24,835 miles, only 0.2% less than the actual value of 24,902 miles. This value remained the most accurate for over a thousand years.
He also proposed the heliocentric theory of gravitation, thus preceding Nicolaus Copernicus in nearly a thousand years.
The 8th century Arabic translation of the Magnum Opus of the Aryabhata, the Aryabhatiya was translated into Latin in the thirteenth century, before the time of Copernicus. By this translation, European mathematicians were able to know the methods for calculating the areas of triangles, volumes of spheres as well as the square and cube roots, while it is also probable that the work of Aryabhata had influence in European astronomy.
The Aryabhata methods of astronomical calculations have been in continuous use for the practice of creating the Panchangam (the Hindu calendar).
One of the books of Aryabhatiya is about mathematics. Aryabhata describes the kuttaka algorithm to solve indeterminate equations. In recent times, this algorithm has also been called the Aryabhata algorithm.
He also created a unique alphabetic code to represent numbers that is now called the Aryabhata cipher.
Aryabhata, and his work Aryabhata-siddhanta, first defined the sine as the modern relationship between half angle and half rope, while also defining the cosine, verse, and inverse sine. His works also contained the oldest tables of sine and verso values (1 - cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of three decimal places. He used the word jya for sine, kojya for cosine, ukramajya for verse, and otkram jya for inverse sine. The words jya and kojyaeventually became sine and cosine respectively after a translation error (see Etymology above). One of the formulas of trigonometry that Aryabhata developed was sin (n + 1) x - sin nx = sin nx - sin (n - 1) x - (1/225) sin nx.