The Pell equation, by Edward Everett Whitford.

THE PELL EQUATION

31

cussed, we must regard the case of those who would show
the dependence of the Hindus on outside sources as not
made out. Dr. Hoernle1 says that Indian arithmetic and
algebra, at least, are entirely of native origin. Strachey2
speaks of the superiority of the Hindus over the Greeks
as being conspicuous in the excellence of their method.
He says, "To maintain that the Bija Ganita rules for the
solution of indeterminate problems might have been had
from any Greek or Arabian, or any modern European
writer [before Bachet de Mezeriac, Fermat, Euler or
Lagrange] would be as absurd as to say that the Newtonian
astronomy might have existed in the time of Ptolemy."
He emphasizes the dissimilarity between the methods of
Diophantus and of the Hindus.3
In addition to the rules of Brahmagupta, Bhaskara
gives some alternative methods, particularly the "cyclical
method," which, as Hankel4 has pointed out, has considerable significance as looking forward to the development
of the modern theory of quadratic forms. The rules of
Bhaskara5 which apply to our equation begin as follows:
"Rules for investigating the square root of a quantity
with additive unity: Let the number be assumed, and be
termed the 'least root.' That number, which, added to,
or subtracted from, the product of its square by the given
coefficient, makes the sum (or difference) give a squareroot, mathematicians denominate a positive or negative
additive; and they call that root the 'greatest' one.
"Having set down the 'least' and 'greatest' roots and
the additive, and having placed under them the same or
others, in the same order, many roots are to be deduced
1 Hoernle, "Indian antiquities," vol. XVII, p. 37, as quoted by G. R.
Kaye, loc. cit.
2E. Strachey, "Bija Ganita; or the algebra of the Hindus," p. 7,
London, 1813.
3 Edinburgh review, vol. XXI, p. 372, Edinburgh, 1813.
4 H. Hankel, p. 200.
5 H. T. Colebrooke, p. 170.