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