Indian Linguistics, Mathematics and Astronomy: Panini to Ramanujan

By Shri J K Bajaj & Shri M D Srinivas

Part 1 / 2: Linguistics

Any study of the Indian tradition of science has to start with linguistics . Not only linguistics is the earliest of Indian sciences to have been rigorously systematised, but also this systematisation became the paradigm example for all other sciences.

Like all sciences and arts of India, Linguistics finds its first expression in the Vedas. For most of the Indian sciences, the elements of study and the categories of analysis were established in the Vaidika period, and the basic data was collected and preliminary systematisation achieved already at that stage. Thus, for the science of Linguistics, we find, in the siksha and pratisakhya texts associated with the various Vedas, a complete and settled list of phonemes appropriately classified into vowels, semi-vowels, sibilants and the five groups of five consonants, all arranged according to the place of articulation that moves systematically from the throat to the lips. Phonetics and phonology are therefore taken for granted by all post-Vaidika authorities on etymology (nirukta) and grammar (vyakarana) , including Yaska and Panini . In the pratisakhya literature we also find the morpho-phonemic (sandhi) rules and much of the methodology basic to the later grammatical literature.

Indian Linguistics finds its rigorous systematisation in Panini’s Ashtadhyayi . The date of this text, like that of much of the early Indian literature, is yet to be settled with certainty. But it is not later than 500 BC. In Ashtadhyayi, Panini achieves a complete characterisation of the Sanskrit language as spoken at his time, and also specifies the way it deviated from the Sanskrit of the Vedas. Using the sutras of Panini, and a list of the root words of the Sanskrit language (dhatupatha) , it is possible to generate all possible valid utterances in Sanskrit. This is of course the main thrust of the generative grammars of today that seek to achieve a grammatical description of language through a formalised set of derivational strings. In fact, till the western scholars began studying generative grammars in the recent past, they failed to understand the significance of Ashtadhyayi: till then Paninian sutras for them were merely artificial and abstruse formulations with little content.

Patanjali (prior to the first century BC) in his elaborate commentary on Ashtadhyayi, Mahabhashya , explains the rationale for the Paninian exercise. According to Mahabhasya, the purpose of grammar is to give an exposition of all valid utterances. An obvious way to do this is to enumerate all valid utterances individually. This is how the celestial teacher Brihaspati would have taught the science of language to the celestial student, Indra. However for ordinary mortals, not having access to celestial intelligence and time, such complete enumeration is of little use. Therefore, it is necessary to lay down widely applicable general rules (utsarga sutras) so that with a comparatively small effort men can learn larger and larger collections of valid utterances. What fails to fit in this set of general rules should, according to the Mahabhashya, then be encompassed in exceptional rules (apavada sutras) , and so on.

In thus characterising grammar, Patanjali expounds perhaps the most essential feature of the Indian scientific effort. Science in India starts with the assumption that truth resides in the real world with all its diversity and complexity. For the Linguist, what is ultimately true is the language as spoken by the people in all their diverse expressions. As Patanjali emphasises, valid utterances are not manufactured by the linguist, but are already established by the practice in the world. One does not go to a linguist for seeking valid utterances, the way one goes to a potter for pots. Linguists make generalisations about the language as spoken in the world. These generalisations are not the truth behind or above the reality of the spoken language. These are not idealisations according to which reality is to be tailored. On the other hand, what is true is what is actually spoken in the real world, and some part of the truth always escapes the grammarians’ idealisation of it. There are always exceptions. It is the business of the scientist to formulate these generalisations, but also at the same time to be always attuned to the reality, to always be conscious of the exceptional nature of each specific instance. This attitude, as we shall have occasion to see, permeates all Indian science, and makes it an exercise quite different from the scientific enterprise of the West.

In Linguistics, after the period of Mahabhashya, grammarians tried to provide continuous refinements and simplifications of Panini. A number of Sanskrit grammars were written. One of them, Siddhanta Kaumudi (c.1600) became eminently successful, perhaps because of its simplicity. These attempts continued till the nineteenth century. Another form of study that became popular amongst the grammarians was what may be called philosophical semantics, where grammarians tried to fix and characterise the meaning of an utterance by analysing it into its basic grammatical components. This, of course, is the major application for which grammar is intended in the first place.

Grammars for other Indian languages were written using Paninian framework as the basis. These grammars were not fully formalised in the sense of Panini. Instead, they started with the Paninian apparatus and specified the transfer rules from Sanskrit and the specific morpho-phonemic (sandhi) rules for the language under consideration. Such grammars for various Prakrit languages of North India, as also of the South Indian languages, continued to be written until the nineteenth century. In the sixteenth century, Krishnadasa even wrote a grammar for the Persian language, Parasi Prakasa , styled on the grammars of the Prakrit languages.K

Part 2 / 2: Astronomy & Mathematics

Among the sciences of the Indian tradition, Astronomy and Mathematics occupy an important place. Indian mathematics finds its early beginnings in the famous Sulabha Sutras of Vaidika literature. Written to facilitate accurate construction of various types of sacrificial altars for the Vaidika ritual, these sutras lay down the basic geometrical properties of plane figures like the triangle, rectangle, rhombus and circle. Basic categories of the Indian astronomical tradition were similarly established in the various Vedanga Jyotisha texts.

Rigorous systematisation of India astronomy begins with the Siddhantas, especially the Brahma or Paitamaha Siddhanta and Surya Siddhanta. Unfortunately, no authentic original versions of these Sidhanta texts are available. The earliest exposition of the Siddhantic tradition is found in the work of Aryabhata (b.476 AD). His Aryabhatiya is a concise text of 121 aphoristic verses containing separate sections on the basic astronomical definitions and parameters; basic mathematical procedures in arithmetic, geometry, algebra and trigonometry; methods of determining mean and true positions of the planets at any given time; and, description of the motion of sun, moon and the planets, along with computations of the solar and lunar eclipses.

Aryabhata was followed by a long series of illustrious astronomers. Some of the well known names are those of Varahamihira (d.578 AD), Brahmagupta (b.598 AD), Bhaskara I (629 AD), Lalla (c.8th century AD), Munjala (932 AD), Sripati (1039 AD), Bhaskara II (b.1114 D), Madhava (c.14th century AD), Paramesvara (c.15th century AD), Nilakantha (c. 16th century AD), Jyeshthadeva (c.16th century AD), Achyuta Pisharoti (c.16th century AD) Ganesa Daivajna (c.16th century AD), Kamalakara (c.17th century AD), Munisvara (c.17th century AD), Putumana Somayaji (c.17th century AD), Jagannatha Pandita (c.18th century AD), and several others.

The texts of several of these astronomers gave rise to a host of commentaries and refinements by later astronomers and become the cornerstones of flourishing schools of astronomy and mathematics. The tradition continued to thrive up to the late eighteenth century. In Kerala and Orissa, original astronomical works continued to be written till much later.

The most striking feature of this long tradition of Indian mathematics and astronomy is the efficacy with which complex mathematical problems were handled and solved. The basic theorems of plane geometry had already been discovered in Sulabha Sutras. Around the time of these Sutras, a sophisticated theory of numbers including the concepts of zero and negative numbers had been established, and simple algorithms for basic arithmetical operations had been formulated using the place-value notation.

By the time of Aryabhatiya, the Indian tradition of mathematics was aware of all the basic mathematical concepts and procedures that are today taught at the high school level. By the 9th or 10th century, sophisticated problems in algebra, such as quadratic indeterminate equations, were solved. By the 14th century, infinite series for trigonometric functions like sine and cosine were derived. By the same time, irrational character of p was recognised, and its value was determined to very high levels of approximation.

The reason for this spectacular success of the Indian mathematicians lies in the explicitly algorithmic and computational nature of Indian mathematics. Indian mathematicians were not trying to discover the ultimate axiomatic truths in mathematics; they were interested in finding methods of solving specific problems that arose in the astronomical and other contexts. Therefore, Indian mathematicians were prepared to work with simple algorithms that may give only approximate solutions to the problem at hand; and they evolved sophisticated theories of error and recursive procedures to keep the approximations in check. This algorithmic methodology persisted in the Indian mathematical consciousness till recently. Srinivasa Ramanujan in the twentieth century seems to have made his impressive mathematical discoveries through the use of this traditional Indian methodology.

Similar pragmatic concerns of determining time and calculating the positions of the various planets and eclipses of the sun and the moon reasonably accurately informed the efforts of the Indian astronomers. In this they were greatly successful. Indian astronomers often take the beginning of the Kaliyuga in 3102 BC as their starting point in their calculations. The Siddhanta texts deal with a much larger period consisting of 4,320,000 years, called a Mahayuga, and sometimes even a period 1000 times greater, called a Kalpa. While working with such long time periods, the Indian astronomers were able to keep their techniques fairly simple and their parameters well refined at all times. Even towards the end of the eighteenth century and early parts of nineteenth, when the astronomical tradition had become dormant in large parts of India, European astronomers were able to locate Brahmins in South India, who could calculate details of the current eclipses to an accuracy comparable to and often better than the best calculations of Europe of the time.

The reasons for the simplicity and accuracy of the Indian astronomical techniques are again to be found in the pragmatic attitude of the Indians towards the sciences. Indian astronomers were in the business to calculate and compute, not to form pictures of the heavens as they ought to be in reality. Indian astronomers do use some geometrical models, but for them these are no more than artefacts to aid their calculations. It is obvious that the astronomical parameters obtained in such a pragmatic approach would get out of tune with reality sooner or later and the calculated positions of the planets would start deviating from actual. Indian astronomers were aware of this and were quite willing to take up the onerous task of continuously observing the skies, continuously checking their computations against observations and repeatedly re-adjusting their parameters so as to make their calculations accord with reality. Thus, the sixteenth century astronomer Nilakantha Somasutvan, finding a contemporary commentator fretting about the circumstance that different Siddhantas mentioned different times, and the computed times differed from the actual ones, exhorts:

O faint-hearted, there is nothing to be despaired of… One has to realise that five Siddhantas had been correct at a particular time. Therefore one has to search for a Siddhanta that does not show discord with the actual observation at the present time. Such accordance has to be ascertained by observers during times of eclipses, etc. When Siddhantas show discord observations should be made with the use of instruments and correct number of revolutions etc. found, and a new Siddhanta enunciated. ” [‘Jyotirmimansa of Nilakantha’, cited in K. V. Sarma and B. V. Subarayappa, A Sourcebook of Indian Astronomy, p.7]

A little later Jyeshthadeva in his Drikkarana recounts how from Aryabhata to the present day the astronomers have adjusted the parameters to accord with observations and how he too is doing the same job for his times.[2] He ends with the advice that ‘henceforth too the deviations that occur should be carefully observed and revisions effected’ .[ ‘Drikkarana of Jyeshthadeva’, cited in K. V. Sarma and B. V. Subarayappa, above, pp.5-6].

Note: This article is borrowed with thanks from “Know Your Bhārat” and you may subscribe to it by sending ‘START’ on WhatsApp to 8884472345. The original source at:

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