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A History of Sanskrit Literature c35.s4: 4. Āryabhaṭa and later Mathematicians

A History of Sanskrit Literature c35.s4

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4. Āryabhaṭa and later Mathematicians

c35.s4.1

Āryabhaṭa, as we have seen, was the first to insert a definitely mathematical⁴ section in his astronomy. He deals in it with evolution and involution, area and volumes; then, after a semi-astronomical section dealing with the circle, shadow problems, &c., he proceeds to progressions and algebraic identities. The rest of the *Gaṇita* deals with examples, save at the close when indeterminate equations of the first degree are taken up (ax+by=c). We find also a remarkably accurate value⁵ of π, viz. 3·1416, and the rule known as the epanthem,⁶ and the type of definition not otherwise in use in India, ‘The product of three equal numbers is a cube and it also has twelve edges.’ On the other hand, we must set clear errors in the volume of a pyramid and a sphere. His notation¹ is unique; it uses the consonants *k* to *m* for 1 to 25, the rest, *y* to *h*, for 30 to 100, while the vowels denote multiplication by powers of 100, *a* being 100⁰ and *au* 100¹⁶.

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⁴ See Kaye, *Indian Mathematics* (1915); *Scientia*, xxv. 1 ff.

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⁵ The epic value is 3·5; Hopkins, *JAOS.* xxiii. 154 f.

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⁶ It is known to Thymaridas (A. D. 380) and Iamblichos (350).

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¹ Cf. Fleet, JRAS. 1911, pp. 109 ff.; IHQ. iii. 116.

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Brahmagupta's work covers very briefly the ordinary arithmetical operations, square 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, cipher, surds, simple algebraic identities, indeterminate equations of the first and second degrees, in considerable detail, and simple equations of the first and second degrees which are briefly treated. Special attention is given to cyclic quadrilaterals. Later, in the ninth century under the Rāṣṭrakūṭa king Amoghavarṣa, we have the *Gaṇitasārasaṁgraha*² of Mahāvīrācārya, which insists on the importance of its subject for every kind of science from cooking to logic, and adapts in its exposition the elegance which is later still further affected by Bhāskara. The work is fuller but rather more elementary than that of Brahmagupta; it gives many examples of solutions of indeterminates, but not the ‘cyclic method’ of Brahmagupta; it introduces geometrical progressions and alone deals, inaccurately, with ellipses, but has no formal algebra. Çrīdhara, born 991, in his *Triçatī*³ is much on the same level as Mahāvīra, but is cited as having dealt with quadratic equations. On Çrīdhara's work and those of Brahmagupta and a certain Padmanābha was based the *Līlāvatī*⁴ of Bhāskara, in which a lovely maiden is addressed, and problems set to her; it includes combinations, while the *Bījagaṇita*, which agrees largely with Brahmagupta, is the fullest and most systematic account of Indian algebra. With Bhāskara ends the active period of Indian mathematics; a school to study his work was founded in 1205 by Caṅgadeva, his grandson,⁵ but its interest seems to have been given to astrology. Of dubious age is the mathematical manuscript known as the Bakhshālī manuscript,¹ from its place of discovery in Peshāwar. It is written in Sūtra style with examples in Çlokas, taken from daily life, and explanations in prose. The mixed Sanskrit in which it is written induced Hoernle to ascribe the work to the third or fourth century A.D., and the manuscript on palaeographic grounds was ascribed by him to the eighth or ninth century, but these conclusions are far from being certain, and the work may date much later.

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² Ed. and trans. M. Raṅgācārya, Madras, 1912.

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³ See N. Rāmānujācārya, *Bibl. Math.*, 1913, pp. 203 ff.

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⁴ See H. T. Colebrooke, *Algebra* (1817); his trans. is ed. H. Ch. Banerji, Calcutta, 1893. Cf. Brockhaus, BSGW. 1852, pp. 1-46.

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⁵ See EI. i. 338 ff.

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¹ Hoernle, OC. VII, i. 128 ff.; IA. xvii. 33 ff. Contrast Kaye, JPASB. 1907, pp. 498 ff.; 1912, pp. 349 ff.

Cites (5)