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Science After Aryabhatta

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In the earliest period of Indian science, it is exceptional when we know the authorship of a text or an idea. For example, although Yajnavalkya and Lagadha describe considerable astronomy, we do not know if this was developed by them or they merely summarized what was then well known. Likewise we are not sure of the individual contributions in the Shulba Sutras--- of Baudhayana, Apastamba, and other authors--- which describe geometry, or in Pingala's Chhandahsutra which shows how to count in a binary manner. The major exception to the anonymous nature of early Indian science is the grammatical tradition starting with Panini. This tradition is an application of the scientific method where the infinite variety of linguistic data is generated by means of a limited number of rules.

With Aryabhata, we enter a new phase in which it becomes easier to trace the authorship of specific ideas. But even here there remain other aspects which are not so well understood. For example, the evolution of Indian medicine is not as well documented as that of Indian mathematics. Neither do we understand well the manner in which the philosophical basis underlying Indian science evolved.

Thus many texts speak of the relativity of time and space---abstract concepts that developed in the scientific context just a hundred years ago. The Puranas speak of countless universes, time flowing at different rates for different observers and so on.

The Mahabharata speaks of an embryo being divided into one hundred parts each becoming, after maturation in a separate pot, a healthy baby; this is how the Kaurava brothers are born. There is also mention of an embryo, conceived in one womb, being transferred to the womb of another woman from where it is born; the transferred embryo is Balarama and this is how he is a brother to Krishna although he was born to Rohini and not to Devaki. There is an ancient mention of space travellers wearing airtight suits in the epic Mahabharata which may be classified as an early form of science fiction.

Universes defined recursively are described in the famous episode of Indra and the ants in Brahmavaivarta Purana. Here Vishnu, in the guise of a boy, explains to Indra that the ants he sees walking on the ground have all been Indras in their own solar systems in different times! These flights of imagination are to be traced to more than a straightforward generalization of the motions of the planets into a cyclic universe. They must be viewed in the background of an amazingly sophisticated tradition of cognitive and analytical thought.

The context of modern science fiction books is clear: it is the liberation of the earlier modes of thought by the revolutionary developments of the 20th century science and technology. But how was science fiction integrated into the mainstream of Indian literary tradition two thousand years ago? What was the intellectual ferment in which such sophisticated ideas arose?

Of the eighteen early siddhantas the summaries of only five are available now. In addition to these siddhantas, practical manuals, astronomical tables, description of instruments, and other miscellaneous writings have also come down to us. The Puranas also have some material on astronomy. Here we just list some of the main names in astronomy after 450 CE.

Aryabhata (born 476) 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 BCE. 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.

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.


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.


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 Somayaji

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.

Source: T.R.N. Rao and S. Kak, Computing Science in Ancient India. USL Press, Lafayette, 1998.

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