The Pell equation, by Edward Everett Whitford.

30

THE PELL EQUATION

which lead to the integral solutions of
x2 - Ay2 = =1 1
are the most important developments of ancient Hindu
mathematics. We can hardly suppose that they originated with Brahmagupta. The very wording of the
problem we have quoted would imply that such problems
were to some extent known and that he expected his
pupils to be successful in their solution. Sometimes he
does not understand the rules he gives, and certain of the
rules are followed by inappropriate examples.1 In several cases he solves part of an example and says, "The
purport of the rest of the question is shown further on."
Again he finds fault with a rule and says, "What occasion
then is there for it?"2 Many other passages3 could be
cited which go to show that he is not the author of the
rules which he gives. Some think that these facts point
to a dependence4 of the Hindus on Greek algebra, but
in view of the early Hindu extraction of square root and
its natural consequences, which have already been dis1 H. T. Colebrooke, p. 366.
2 H. T. Colebrooke, ~ 64.
3 G. R. Kaye, "Sources of Hindu mathematics," Journal of the Royal
Asiatic Society, London, July, 1910. H. G. Zeuthen, "Histoire des
mathematiques," p. 239.
4 G. R. Kaye, "Notes on Indian mathematics-Arithmetical notation,"
Journal and Proceedings of the Asiatic Society of Bengal, vol. III, No. 7,
p. 501, Calcutta, 1907. Kaye says: "The only resemblance between the
matter of the Bakhshali manuscript and Brahmagupta's work that Dr.
Hoernle points out lies in the fiftieth sutra of the MS. and chapter XVIII,
~ 84 of Brahmagupta's algebra. Peculiar significance attaches to this
problem, for it was fully dealt with by Diophantus and fully expounded in
the algebra of Alkarkhi which was based on that of Diophantus. The
problem in the Bakhshali MS. expressed in modern notation is
2 + 5 = m2, x2- 7 = n2,
and is based on the fact that 1/4((5 + 7)/2 + 2)2 + (5 + 7) is a perfect
square. This formula is given by Alkarkht. This rather remarkable
coincidence unmistakably points to Diophantus as one of the ultimate
sources of both Brahmagupta's work and the Bakhshali arithmetic."