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

LEMMAS OF BRAHMAGUPTA 247
quent operations of their composition are performed. Secondly,
it may also mean that the earlier operations are made with
one optionally chosen number and one interpolator, and the
subsequent ones are carried out after the repeated statement of
those roots for the second time. It is also implied that in the
composition of the quadratic roots, their products may be added
together or subtracted from each other.
In other words, if x=a, y=β be a solution of the equation :
Nx² + k = y²,
and x=a', y=β' be a solution of
Nx² + k' = y²,
then according to the above
x = aβ' ± a'β, y = ββ' ± Naa'
is a solution of the equation
Nx² + kk' = y².
In other words, if
Na² + k = β²
Na'² + k' = β'²
then
· N(aβ' ± a'β)² + kk' = (ββ' ± Na')² (I)
In particular, taking a=a', β=β' and k=k', Brahmagupta
finds from a solution x=a, y=β of the equation
Nx² + k² = y²
a solution x=2aβ, y=β² + Na² of the equation
Nx² + k = y²
That is, if
Na² + k = β²
then
N(2aβ)² + k² = (β² + Na²)² (II)
This result will be hereafter called Brahmagupta's Corol-
lary.
Thus Brahmagupta's First Lemma says that if two solutions
of the equation (of the Square-nature) Nx² + 1 = y² are known,
then any number of other solutions can be found. For example
if two solutions of the Square-nature are (a, b) and also (a', b'),
then two other solutions will be :
x = ab' ± a'b, y = bb' ± Naa'.