Hindu Achievements in Exact Science: A Study in the History of Scientific Development c8: II. Algebra

Hindu Achievements in Exact Science: A Study in the History of Scientific Development c8

Language: English

II. Algebra

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ALGEBRA is a Hindu science in spite of the Arabic name. Cajori suspects that Diophantus (A.D. 360), the first Greek algebraist, got the first glimpses of algebraic knowledge from India. According to Heath, the Europeans were anticipated by the Hindus in the symbolic form of algebra. According to De Morgan, the work of Diophantus is hardly algebraic in the sense in which that term can be applied to the science of India.³³ According to Hankel, the Hindus are the real inventors of algebra if we define algebra as the application of arithmetical operations to both rational and irrational numbers or magnitudes.⁴

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The mathematician who systematized the earlier algebraic knowledge of the Hindus and thus became the founder of a new science is Aryabhata (born A.D. 476 at Pataliputra on the Ganges in Eastern India). He was thus over a century later than Diophantus; but Smith proves that neither in methods nor in achievements could the Greek be the inspirer of the Hindu.³³

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The points in which the Hindu algebra appears particularly distinguished from the Greek are thus enumerated by Colebrooke:

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1. A better and more comprehensive algorithm.
2. The management of equations involving more than one unknown term. (This adds to the two classes noticed by the Saracens, viz., simple and compound.)
3. The resolution of equations of a higher order, in which if they achieved little, they had at least the merit of the attempt, and anticipated a modern discovery in the solution of biquadratics.
4. General methods for the solution of indeterminate problems of first and second degrees, in which they went far beyond Diophantus, and anticipated the discoveries of modern algebraists.
5. Application of algebra to astronomical investigation and geometrical demonstration, in which also they hit upon some methods which have been reinvented in later times.

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It was thus not a "primitive" algebra that the Hindus developed. The achievements of Indian algebra from the fifth to twelfth century A.D. have in some cases anticipated the discoveries of the seventeenth and eighteenth centuries in Europe. Modern algebraists have thus only re-discovered the already known truths.

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The Hindu algebra of this period was the principal feeder of Saracen algebra through Yakub and Musa, and indirectly influenced to a certain extent mediæval European mathematics. It may have fostered the development of mathematics in China also, and through that, of Japan. According to Williams, the Hindu processes in algebra were known to the mathematicians of the Chinese empire, "and are still studied" in the Middle Kingdom "though all intellectual intercourse between the two countries has long ceased."

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The progress of Hindu algebra (mainly in Southern India) after Bhaskara (twelfth century) was, as Seal suggests, parallel to the development in China and Japan. But this is a subject that awaits further research.

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The Hindu discoveries in algebra may be thus summarized from the recent investigations of Nalinbehari Mitra:

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1. The idea of an absolutely negative quantity.
2. The first exposition of the complete solution of the quadratic equation (Brahmagupta A.D 598–660)
3. Rules for finding permutations and combinations (Bhaskara, born 1114). These were unknown to the Greeks.
4. Indeterminate equations: "The glory of having invented general methods in this most subtle branch of mathematics belongs to the Indians." ⁴
5. Indeterminate equations of the second degree.

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In the light of Comparative Chronology, these discoveries are remarkable evidences of the fecundity of the Hindu brain in "*exact*" science. The three great anticipations of modern algebra are enumerated and appreciated by Colebrooke in the following terms:

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1. The demonstration of the noted proposition of Pythagoras concerning the square of the base of a rectangular triangle, equal to the squares of the two legs containing a right angle. The demonstration is given in two ways in Bhaskara’s algebra (twelfth century). The first of them is the same which is delivered by Wallis (1616–1703) in his treatise on angular sections, and as far as appears, then given for the first time.
2. The general solution of indeterminate problems of the first degree. It was first given among moderns by Bachet de Meziriac in 1624.
3. Solution of indeterminate problems of the second degree . . . a discovery which among the moderns was reserved for Euler (1707–83). To him among the moderns we owe the remark which the Hindus had made more than a thousand years ago, that the problem was requisite to find all the possible solutions of equations of this sort.

Linked texts (1)

Bījagaṇita — Bhāskara II, Sanskrit