History of Hindu Mathematics: A Source Book v2.c17.s14.56

v2.c17.s14.56

**General Solution of the Square-Nature.** It is clear from Brahmagupta's Lemma (I) that when two solutions of the Square-nature, $Nx²+1=y²,$ are known, any number of other solutions can be found. For, if the two solutions be (a, b) and (a', b'), then two other solutions will be $x=ab'± a'b, y=bb'± Naa'.$ Again, composing this solution with the previous ones, we shall get other solutions. Further, it follows from Brahmagupta's Corollary that if (a, b) be a solution of the equation, another solution of it is (2ab, b²+Na²). Hence, in order to obtain a set of solutions of the Square-nature it is necessary to obtain only one solution of it. For, after having obtained that, an infinite number of other solutions can be found by the repeated application of the Principle of Composition. Thus Śrîpati (1039) observes : “There will be an infinite (set of two roots).”¹ Bhâskara II (1150) remarks : “Here (*i.e.*, in the solution of the Square-nature) the roots are infinite by virtue of (the infinitely repeated application of) the Principle of Composition as well as of (the infinite variety of) the optional values (of the first roots).”² Nârâyaṇa (1350) writes, “By the Principle of Composition of equal as well as unequal sets of roots, (will be obtained) an infinite number of roots.”³