A History of Sanskrit Literature c35.s5: 5. Greece and Indian Mathematics

A History of Sanskrit Literature c35.s5

Language: English

5. Greece and Indian Mathematics

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The relation of India to Greek mathematics in this period is one of complexity and difficulty, and it cannot be disposed of by insisting on the indebtedness of India to Greece in respect either of astronomy or astrology, for in both cases the exact extent of that influence is obscure.² The question is obscured also by the fact that we have lost the works of Hypatia, murdered in 415 by the Alexandrian mob, and therefore cannot trace the progress of mathematics after Diophantos (*c.* 260). The visit of the philosophers expelled from the schools of Philosophy in Athens in 530 to the court of Chosrau of Persia in 532 was brief, and it is not much use speculating on its possibilities, though Damaskios and Simplikios, who had some repute in mathematics, were among those involved. The facts are that, as regards indeterminate equations, the Greeks by the fourth century had achieved rational solutions, not necessarily integral, of equations of the first and second degree and of some cases of the third degree. The Indian records go distinctly beyond this. Brahmagupta shows a complete grasp of the integral solution of ax ± by = c, and he indicates one method, called by Bhāskara the method by composition, of the solution of Du² + 1 = t². Bhāskara adds the cyclic method, as he calls it, and the combination of these two methods, which gives integral solutions, has been styled by Hankel the finest thing achieved in the theory of numbers before Lagrange. To find an ultimate Greek origin for these discoveries seems due rather to a *parti pris* than to justice.

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² Kaye (*Hindu Mathematics*) goes rather far in his claims for Greece. Contrast D. E. Smith in Raṅgācārya's *Gaṇitasārasaṁgraha*, pp. xxi ff. For older views see Hankel, *Gesch. der Math.* (1874), pp. 172 ff.; Cantor, *Gesch. der Math.*, i. 505 ff.; M. Simon, *Gesch. der Math.* (1909). See also J. L. Heiberg, *Mathematics and Physical Science in Classical Antiquity* (1922); D. E. Smith, *Hist. of Mathematics* (1925); Peet, *The Rhind Mathematical Papyrus* (1923); Heath, *Hist.* (1921).

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Another point on which special interest in India was centred was the question of obtaining integral solutions in the case of the rational right-angled triangle. The results achieved there are interesting and may be compared with similar work, not by any means identical, of Euclid and Diophantos, as well as solutions ascribed to Plato by Proclus. Brahmagupta, Mahāvīra, and Bhāskara all contribute to the topic, and the former first states certain historically interesting problems; the sum of the sides is 40 and area 60; the sum of the sides is 56 and the product 7 × 600; the area is numerically equal to the hypotenuse; or to the product of the sides. Brahmagupta further did important work on cyclic quadrilaterals, achieving as one of his results his theorem: x² = (ad + bc) (ac + bd) / (ab + cd), and y² = (ab + cd) (ac + bd) / (ad + bc), where x and y are the diagonals of the cyclic quadrilateral, a, b, c, d. Mahāvīra and Çrīdhara repeat some of his matter, but their commentators show ignorance of the principle, and Bhāskara severely condemns both him who puts such a question and him who answers it. It is interesting, but by no means a proof of borrowing, that a commentator on Brahmagupta constructs from triangles new triangles and actually uses the same examples as Diophantos. Nor can we draw any definite conclusion from the fact that Indian mathematics in regard to geometry shows an absence of definitions, does not deal with angles nor mention parallels, nor give a theory of proportion, while traditional inaccuracies are common and knowledge is in the later period steadily declining. The same facts are seen in Greek geometry from A.D. 300, and possibly we can best understand Indian facts as indicating borrowings from such a decadent school, but there is no cogency in the contention.

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The independence and originality of Indian mathematics have been defended on the score that the love of dealing with large numbers and making calculations is recorded early for India, where it is alleged the abacus¹ was invented, and that the numerals of the west are borrowed from India, where the place value system¹ was introduced. The abacus, on the other hand, is asserted to be of comparatively modern date, and it has been suggested that India borrowed it from Greece. The question of numbers is very dubious; the figures of the Brāhmī or Kharoṣṭhī notation have not place value, but their origin is uncertain. India knows, beside the unique system of Āryabhaṭa, the use of words for numbers, and place value is actually found in inscriptions from the ninth century onwards, but its earliest occurrence on an inscription of 595 is doubted, though the *Yogabhāṣya* clearly knows it, as did Āryabhaṭa and Varāhamihira.² But there is considerable antiquity for the tradition of the borrowing; the Indian figures were known in Syria in A.D. 662, and Mascūdi ascribes the origin to a congress of sages gathered together by king Brahma. The probability still remains that India did render a great service in this regard, and in any case excelled Greece. It is, of course, perfectly possible, and in view of the facts as regards astronomy and astrology not at all unlikely, that India borrowed its impulse to mathematics from Greece in the shape of those manuals whence she borrowed her astronomy, and this is certainly supported by the fact of Āryabhaṭa's evaluation of π, which is also ascribed to Puliça, and it was known to Apollonios and Ptolemy.

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¹ *Contra*, Kaye, JPASB. 1908, pp. 293 ff., but see Fleet, JRAS. 1911, pp. 121, 518 ff. *Cf.* IHQ. iii. 357 ff.

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¹ See Kaye, JPASB. 1907, pp. 475 ff.; Bubnow, *Arithmetische Selbständigkeit der europäischen Kultur* (1914); *contra*, D. E. Smith and L. C. Karpinski, *The Hindu Arabic Numerals* (1911); Nau, JA. sér. 10, xvi. 225-7; C. de Vaux, *Scientia*, 1917, pp. 273 f.; Sukumar Ranjan Das, IHQ. iii. 100 ff., 356 ff.

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² Woods, HOS. xvii. 216.

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Recently the claim of India to have inspired Arabic mathematics has also been attacked, on the score that Muḥammad ibn Mūsā (782) in his Algebra is not, as was long believed, really under Indian but under Greek influence, and a good case seems to have been made against any substantial importance of India in this regard; but there seems equally slight ground for the counterclaim that India borrowed after Brahmagupta from Arab mathematics at least in the period up to Bhāskara. But it must be remembered that Arabian science from A.D. 771 borrowed freely from Indian astronomy,³ translating and adapting both Āryabhaṭa and Brahmagupta, so that, if we hold that Arabia was independent in mathematics of India, we must recognize that borrowing of astronomy or astrology must not be adduced as conclusive for borrowing of mathematics. Coincidences with Chinese¹ mathematics are numerous and interesting, and it has, of course, long been urged that China invented the system of Nakṣatras found in early Indian astronomy,² but at present at least the case for dependence on China is not made out, and Indian influence on China is proved sufficiently by the history of Chinese Buddhism and the discoveries in Central Asia.

Linked texts (1)

Kauṭilīya Arthaśāstra — Kauṭilya, Sanskrit