SECOND LEMMA OF BRAHMAGUPTA This rule may be expressed in terms of symbols as follows. Suppose the Varga-prakrti (Square-nature) to be Nx²p²d=y². so that its interpolator (kşepa) p'd is exactly divisible by the square p. Then, putting therein u=x/p, v=y/p, we derive the equation Nu²td=v² whose interpolator is equal to that of the original Square-nature divided by pª. It is clear that the roots of the original equation are p times those of the derived equation. Indian algebraists have usually suggested the following method to obtain a first solution of Nx²+1=y²: OF 251 Take an arbitrary small rational number, a, such that its square multiplied by the gunaka 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 Na²±k-² where a, k, and ß are rational numbers. Let us call this relation as the Auxiliary Equation. Then by Brahmagupta's Coro- Ilary, we get from it the relation Rational Solution N(2a8)+k² (3² +Na²)³, 9 Na² N(²) ² + 1 =(²²+₁²) ² k 2a ħ Hence, one rational solution of the equation Nx²+1=y² is given by và ß² +Na² k Work on the rational solution of the Square-nature has been also done by Sripati. 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:
