The Pell equation, by Edward Everett Whitford.

32

THE PELL EQUATION

from them by composition. Wherefore their composition
is propounded.
"The 'greatest' and 'least' roots are to be reciprocally
multiplied crosswise; and the sum of the products to be
taken for a 'least' root. The product of the two (original)
'least' roots being multiplied by the given coefficient,
and the product of the 'greatest' roots being added thereto, the sum is the corresponding greatest root; and the
product of the additives will be the [new] additive.
"Or the difference of the products of the multiplication
crosswise of the greatest and least roots may be taken for
a 'least' root; and the difference between the product of
the two [original] least roots multiplied together and
taken into the coefficient, and the product of the greatest
roots multiplied together, will be the corresponding
'greatest' root: and here also the additive will be the
product of the two [original] additives.
"Let the additive divided by the square of an assumed
number be a new additive; and the roots, divided by that
assumed number, will be the corresponding roots. Or the
additive multiplied [by the square], the roots must, in
like manner, be multiplied [by the number put].
"Or divide the double of an assumed number by the
difference between the square of that assumed number and
the given coefficient; and let the quotient be taken for
the 'least' root, when one is the additive quantity; and
from that find the 'greatest' root."'
The commentator Crishna-bhatta has demonstrated
these rules in a cumbersome manner in which he has
used for symbols the first syllables of some of the words
involved. Colebrooke has abbreviated this, and in
1Lord Brouncker's first solution,
2r 2 r2 + A )2
A ~r~A) +-l=(r-7-__
shows a marked resemblance to this rule. This was before he began to
seek for integral solutions.