A History of Sanskrit Literature c35.s5.3

c35.s5.3

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.