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

252 BRAHMAGUPTA AS AN ALGEBRAIST
Unity is the lesser root. Its square multiplied by the
prakṛti is increased or decreased by the prakṛti com-
bined with an (optional) number whose square-root
will be the greater root. From them will be obtained
two roots by the Principle of Composition¹
Thus if m² be the rational number optionally chosen, one
shall have the identity :
N.1²+(m²—N)=m²,
or N.1²—(N—m²)=m²
Then by applying Brahmagupta's Corollary we get
N(2m)²+(m² ∼ N)²=(m²+N)²
∴ N (2m / (m² ∼ N))² + 1 = ((m² + N) / (m² ∼ N))²
Hence
x = 2m / (m² ∼ N), y = (m² + N) / (m² ∼ N)
where m is any rational number, is a solution of the equation
Nx² + 1 = y².
This rational solution of the varga-prakṛti which was used
by Śrīpati in 1039 A.D. was rediscovered in Europe by Broun-
cker in 1657.
We shall close this discussion by taking an illustration
from Bhāskara II :
Problem : Tell me, O mathematician, what is that square
which multiplied by 8 becomes, together with
unity, a square; and what square multiplied by
11 and increased by unity, becomes a square.
This means that we have to solve the equations :
8x² + 1 = y² ......(i)
11x² + 1 = y² ......(ii)
In the second example, let us assume 1 as the lesser root.
Following the method of Śrīpati, let us multiply its square by
the prakṛti (here in eq. ii, prakṛti is 11), then let us subtract 2
(an optional number) and then extracting the square-roots we

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1. Śrīpati, Siddhānta-śekhara XIV. 33