Indian Science and Technology in the Eighteenth Century: Some Contemporary European Accounts c13.59

c13.59

Bhascara's second method (*Vij.* § 80-81) for a solution of the problem on which all indeterminate ones of this degree depend, is exactly the same, which Lord Brouncker devised to answer a question proposed by way of challenge by Fermat in 1657. The thing required was a general rule for finding the innumerable square numbers, which multiplied by a proposed (non-quadrate) number, and then assuming an unit, will make a square. Lord Brouncker's rule, putting n for any given number, r² for any square taken at pleasure, and d for difference between n and r²(r² ~ n) was (4r²)/(d²) = (2r/d × 2r/d) the square required. In the Hindu rule, using the same symbols, 2r/d is the square root required.⁸⁴ But neither Brouncker, nor Wallis, who himself contrived another method, nor Fermat, by whom the question was proposed, but whose mode of solution was never made known by him, (probably because he had not found anything better than Wallis and Brouncker discovered)⁸⁵ nor Frenicle, who treated the subject without, however, adding to what had been done by Wallis and Brouncker,⁸⁶ appear to have been aware of the importance of the problem and its universal use: a discovery, which, among the moderns, was reserved for Euler in the middle of the last century. To him, among the moderns, we owe the remark, which the Hindus had made more than a thousand years before,⁸⁷ that the problem was requisite to find all the possible solutions of equations of this sort. La Grange takes credit for having further advanced the progress of this branch of the indeterminate analysis, so lately as 1767;⁸⁸ and his complete solution of equations of the 2nd degree appeared no earlier than 1769.⁸⁹