ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

SECOND LEMMA OF BRAHMAGUPTA 251
This rule may be expressed in terms of symbols as follows.
Suppose the Varga-prakṛti (Square-nature) to be
Nx² ± p²d = y²,
so that its interpolator (kṣepa) p²d is exactly divisible by the
square p². Then, putting therein u = x/p, v = y/p, we derive the
equation
Nu² ± d = v²
whose interpolator is equal to that of the original Square-nature
divided by p². It is clear that the roots of the original equation
are p times those of the derived equation.
Rational Solution
Indian algebraists have usually suggested the following
method to obtain a first solution of Nx² + 1 = y² :
Take an arbitrary small rational number, α, such that
its square multiplied by the guṇaka N and increased
or diminished by a suitably chosen rational number k
will be an exact square.
In other words, we shall have to obtain empirically a rela-
tion of the form
Nα² ± k = β²
where α, k, and β are rational numbers. Let us call this
relation as the Auxiliary Equation. Then by Brahmagupta's Coro-
llary, we get from it the relation
N(2αβ)² + k² = (β² + Nα²)²,
or N(2α β / k)² + 1 = ((β² + Nα²) / k)²
Hence, one rational solution of the equation Nx² + 1 = y² is
given by
x = 2αβ / k , y = (β² + Nα²) / k
Work on the rational solution of the Square-nature has been
also done by Śrīpati. In fact, his solution, given in 1039 A.D.
is of historical significance. He derives the rational solution
without the aid of the "auxiliary equation." He gives the follo-
wing rule :