The Pell equation, by Edward Everett Whitford.

34

THE PELL EQUATION

they called "cyclic." The commentator Surin explaining
the name cyclic uses these words, "Finding from the roots
a multiplier and quotient; and thence new roots; whence
again a multiplier and a quotient, and roots from them;
and on in a continued round."
"Rule' for the cyclic method: Making the 'least' and
'greatest' roots and additive, a dividend additive and
divisor, let the multiplier be thence found. The square
of that multiplier being subtracted from the given coefficient, or this coefficient being subtracted from that square
(so as the remainder be small); the remainder, divided
by the original additive, is a new additive; which is reversed if the subtraction be [of the square] from the
coefficient. The quotient corresponding to the multiplier
[and found with it] will be the 'least' root: whence the
' greatest' root may be deduced. With these the operation
is repeated, setting aside the former roots and additive.
This mathematicians call that of the circle. Thus are
integral roots found with four, two or one [or other
number] for additive: and composition serves to deduce
roots for additive unity, from those which answer to the
additives four and two [or other number]."
The words "or other number" were evidently inserted
in the rule by some commentator or by the translator
and might well be omitted. The cyclic method is to be
continued until the right member of
x2 - Ay2 = s
becomes - 4 or - 2 or - 1, and then a single step which
Bhaskara calls composition brings us to the solution of
the equation
x2 - Ay2 = 1.
The steps in the cyclic method of solution of the equation
(1) ay2 + 1 = x2
1H. T. Colebrooke, p. 175.