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'.