India’s romance with numbers can be seen in the mention of large numbers up to 1019 in the Black Yajurveda. The sources for mathematical computing in India in the early phase are the Vedic books including the Brāhmana texts, the Śulbasūtras (texts on altar geometry), Jyotisha (astronomy), the Chandahśāstra (Pingala’s book on prosody), Pānini’s grammar, the Nātya Śāstra (Bharata Muni’s text on music, drama, and dance), and nonmathematical texts such as the Mahābhārata and diverse śāstras (scientific and philosophical texts).
The Indian approach to the world was classificatory and computational. In astronomy, an attempt was made to reconcile the motions of the sun, the moon, and the planets that required periodic corrections. In grammar, the most economical rules for different constructions were sought. Not only the sign for zero, but also the binary number system, infinity and its operations, the ideas of metarules, algebraic transformation, recursion, hashing, mathematical logic, formal grammars, and high level language description arose first in India.
The Śulbasūtras belong to the Vedāngas, or supplementary texts of the Vedas that deal with construction of geometric altars. Their contents, written in the condensed sūtra style, cover geometrical propositions and problems related to rectilinear figures and their combinations and transformations, squaring the circle, as well as arithmetical and algebraic solutions to these problems. The root śulb means measurement, and the word śulba means a cord, rope, or string.
The extant Śulbasūtras belong to the schools of the Yajurveda. The most important Śulba texts are the ones by Baudhāyana, Āpastamba, Kātyāyana, and Mānava. They have been generally assigned to the period 800–500 BCE of which Baudhāyana’s text is the oldest. Baudhāyana begins with units of linear measurement and then presents the geometry of rectilinear figures, triangles, and circles, and their transformations from one type to another using differences and combinations of areas.Approximations to the square root of 2 and to π are given next.
In the methods of constructing squares and rectangles, several examples of Pythagorean triples are provided. It is clear from the constructions that both the algebraic and the geometric aspects of what we call the Pythagorean theorem were known. Several geometric constructions in these texts are based on algebraic solutions of simultaneous equations of linear and quadratic types. It appears that geometric techniques were often used to solve algebraic problems.
Jyotisha Recent researches have established that the altar geometry of the Śulbas was used to represent astronomical facts related to the knowledge of the lunar and the solar years. The solution to these problems involved solution to indeterminate linear equations. Lagadha’s Vedānga Jyotisha (1300 BCE), a book on the motions of the sun and the moon, presupposes knowledge of such equations.
Pa¯ nini’s Grammar
Pānini’s Ashtādhyāyī, “The Eight Chapters” (fifth century BCE), provides 4,000 rules that describe the Sanskrit of his day completely. The great variety of linguistic ideas used in the text mirrors the complexity of cognitive relationships that is the secret of its power and success. It is remarkable that Pānini set out to describe the entire grammar in terms of a finite number of rules. Scholars have shown that the grammar of Pānini represents a universal grammatical and computing system.
The Ashtādhyāyī ostensibly deals with the Sanskrit language. However, it presents the framework for a universal grammar that can apply to any language. Two important early commentaries on this grammar are by Kātyāyana and Patanjali. Bhartrihari examined its philosophical basis in an important work in the fifth century AD.
Pānini’s grammar begins with metarules, or rules about rules using a special technical language, or Computing Science in Ancient India 199 C metalanguage. Several sections follow on how to generate words and sentences starting from roots, as well as rules on transformation of structure. The last part of the grammar is a one-directional string of rules, where a given rule in the sequence ignores all rules that follow. Pānini also uses recursion by allowing elements of earlier rules to recur in later rules. This anticipates in form and spirit by more than 2,500 years the idea of a computer program.
In Pānini’s system, a finite set of rules is enough to generate an infinity of sentences. The algebraic structure of Pānini’s rules was not appreciated in the West until about 50 years ago when Noam Chomsky proposed a similar generative structure. Before this, in the nineteenth century, Pānini’s analysis of roots and suffixes and his recognition of ablaut led to the founding of the subjects of comparative and historical linguistics.
Pānini took the idea of action as defined by the verb and developed a comprehensive theory by providing a context for action in terms of its relations to agents and situations. This theory is called the kāraka theory.
These kārakas are: 1. That which is fixed when departure takes place 2. The recipient of the object 3. The instrument, or the main cause of the effect 4. The basis, or location 5. What the agent seeks to attain, deed, object 6. The agent These kārakas do not always correspond to the nature of action; therefore, the kāraka theory is only a via media between grammar and reality. It is general enough to subsume a large number of cases, and where not directly applicable, the essence of the action/ transaction can still be cast in the kāraka mold. To do this, Pānini requires that the intent of the speaker be considered. Rather than a structure based on conventions regarding how to string together words, Pānini’s system is based on meaning.
Chhandahs´a¯ stra and Na¯ tya S ´ a¯ stra
The Chhandahśāstra of Pingala (400 BCE according to tradition) describes the binary number system. The idea of mathematical zero is also implicit in some of Pānini’s rules. The Nātya Śāstra presents important results on permutations and combinations. Further results are to be found in later books on musicology such as the Brihaddeśi and the Sangīta-Ratnākara.
The Siddha¯ ntic Age
Āryabhat.a (born 476) presented the first general solution to the linear indeterminate equation using the method of kuttaka. Brahmagupta (seventh century) solved the quadratic indeterminate equation Nx2 þ 1 ¼ y2.
Brahmagupta’s expressions for the diagonals of a cyclid quadrilateral are considered most extraordinary.
Virahānka (eight century) gave an explicit rule for what we call the Fibonacci sequence. Vīrasena (ninth century) knew the logarithm to base 2 and its properties.
Jayadeva of the same period developed the chakravāla (cyclic) method which, according to Hankel, is the “finest thing achieved in the theory of numbers before Lagrange (18th century).” Bhāskara (born 1114) made further advances to trigonometry and analysis.
Navya Nya¯ ya
Navya Nyāya (New Logic) began in Mithila under the leadership of Gangeśa Upādhyāya (twelfth century).
The scholars of this school developed an elaborate technical vocabulary and logical apparatus that came to be used by philosophers and writers on law, poetics, aesthetics, and ritualistic liturgy. In their technique of analyzing knowledge, judgmental knowledge was analyzed into three kinds of epistemological entities in their interrelations: “qualifiers”; “qualificandum,” or that which must be qualified; and “relatedness.” Further abstract entities used were qualifierness, qualificandumness, and relatedness. The knowledge expressed by the judgment “This is a blue pot” may then be analyzed into the following form: The knowledge that has a qualificandumness in what is denoted by ‘this’ is conditioned by a qualifierness in blue and also conditioned by another qualifierness in potness.
At its zenith during the time of Raghunātha (1475– 1550), this school developed a methodology for a precise semantic analysis of language. Its formulations are equivalent to mathematical logic.
The Kerala school of mathematics flourished during 1200–1600 in South India. One of its great figures, Mādhava (ca. 1340–1425), also provided methods to estimate the motions of the planets. He gave power series expansions for trigonometric functions, and for π correct to 11 decimal places.
A very prolific scholar who wrote several books on astronomy, Nīlakant.h.a (ca. 1444–1545), found the correct formulation for the equation of the center of the planets and his model is nearly a true heliocentric model of the solar system. In it, the planets go in eccentric orbits around the mean Sun, which in turn goes around the Earth. He also improved upon the power series techniques of Mādhava.
These mathematicians made truly fundamental advances in analysis including infinite series and calculus. Almeida and his collaborators have suggested 200 Computing Science in Ancient India that Jesuit missionaries brought this work to Europe and it may have provided the spark for the European mathematical revolution.
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