August 29, 2015 by Indian Schools of Thought
Aryabhatta, born in 476 A.D., in Pataliputra was one of the greatest scientists that India had produced. He was the first to treat Mathematics as a distinct subject and his work Aryabhatiyam dealt with evolution and involution, area and volume, progression and algebraic identities and indeterminate equations of the first degree. In the realm of Geometry, the work describes several properties of the circle, discusses questions connected with projective geometry and give a value for pai, far accurate than any suggested till then. That Trigonometry was also being cultivated at this time will become clear from the use of the sine functions made for solving the problems of astronomy. In the realm of Astronomy, Aryabhatta’s work Surya Siddhanta examines and explains the true causes of the solar and lunar eclipses. He was the first to hold the view that eclipses were caused by the shadow of the earth falling on the moon. His calculation of the size of the earth is very near that figure which is estimated by modern astronomers. He was the first Indian astronomer t discover and declared that the earth rotates round its axis and he was the first to discover sine functions and utilize them in astronomy.
Aryabhata is the author of the first of the later siddhantas called Aryabhatiyam which sketches his mathematical, planetary, and cosmic theories. This book is divided into four chapters: (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 epicycles, (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses. The parameters of Aryabhatiyam have, as their origin, the commencement of Kaliyuga on Friday, 18th February, 3102 B.C.E. He wrote another book where the epoch is a bit different.Aryabhata took the earth to spin on its axis; this idea appears to have been his innovation. He also considered the heavenly motions to go through a cycle of 4.32 billion years; here he went with an older tradition, but he introduced a new scheme of subdivisions within this great cycle. According to the historian Hugh Thurston, “Not only did Aryabhata believe that the earth rotates, but there are glimmerings in his system (and other similar systems) of a possible underlying theory in which the earth (and the planets) orbits the sun, rather than the sun orbiting the earth. The evidence is that the basic planetary periods are relative to the sun.” That Aryabhata was aware of the relativity of motion is clear from this passage in his book,“Just as a man in a boat sees the trees on the bank move in the opposite direction, so an observer on the equator sees the stationary stars as moving precisely toward the west.”
Varahamihira (died 587) lived in Ujjain and he wrote three important books: Panchasiddhantika, Brihat Samhita, and Brihat Jataka. The first is a summary of five early astronomical systems including the Surya Siddhanta. (Incidently, the modern Surya Siddhanta is different in many details from this ancient one.) Another system described by him, the Paitamaha Siddhanta, appears to have many similarities with the ancient Vedanga Jyotisha of Lagadha. Brihat Samhita is a compilataion of an assortment of topics that provides interesting details of the beliefs of those times. Brihat Jataka is a book on astrology which appears to be considerably influenced by Greek astrology.
Varahamihira who flourished in the close of 5th century A.D. was another famous astronomer and mathematician. His work Brihatsamhita is an encyclopedia of useful information in several branches of knowledge such as astronomy, physical geography, botany, architecture, sculpture, movements of heavenly bodies and their effect upon men, etc. Historians will remain ever grateful to him for his Panchasiddhantika, which gives a concise account of the five Siddhantas i.e., astronomical works viz., Paitamaha, Romaka, Paulisa, Vasishta and Surya, that were in use in India during the 3rd and 4th century A.D.
Brahmagupta of Bhilamala in Rajasthan, who was born in 598, wrote his masterpiece, Brahmasphuta Siddhanta, in 628. His school, which was a rival to that of Aryabhata, has been very influential in western and northern India. Brahmagupta’s work was translated into Arabic in 771 or 773 at Baghdad and it became famous in the Arabic world as Sindhind. One of Brahmagupta’s chief contributions is the solution of a certain second order indeterminate equation which is of great significance in number theory. Another of his books, the Khandakhadyaka, remained a popular handbook for astronomical computations for centuries. Another famous astronomer and mathematician of ancient India was Brahmagupta.
Long before Newton, he declared the Law of Gravity. His works Brahmasiddhanta, Khandakhadya and Dhyanagraha covers arithmetical operations, squares and cube roots, rule of three interest, progressions, geometry, including treatment of the rational right angled triangle and the elements of the circle, elementary mensuration of solids, shadow problems, negative and positive quantities, ciphers, surds, etc.
Bhaskara (born 1114), who was from the Karnataka region, was an outstanding mathematician and astronomer. Amongst his mathematical contributions is the concept of differentials. He was the author of Siddhanta Shiromani, a book in four parts: (i) Lilavati on arithmetic, (ii) Bijaganita on algebra, (iii) Ganitadhyaya, (iv) Goladhyaya on astronomy. He epicyclic eccentric theories of planetary motions are more developed than in the earlier siddhantas. Subsequent to Bhaskara we see a flourishing tradition of mathematics and astronomy in Kerala which saw itself as a successor to the school of Aryabhata. We know of the contributions of very many scholars in this tradition, of whom we will speak only of two below.
Madhava (c. 1340-1425) developed a procedure to determine the positions of the moon every 36 minutes. He also provided methods to estimate the motions of the planets. He gave power series expansions for trigonometric functions, and for pi correct to eleven decimal places.
Nilakantha (c. 1444-1545) was a very prolific scholar who wrote several works on astronomy. It appears that Nilakantha found the correct formulation for the equation of the center of the planets and his model must be considered a true heliocentric model of the solar system. He also improved upon the power series techniques of Madhava. The methods developed by the Kerala mathematicians were far ahead of the European mathematics of the day.