BRAHMAGUPTA AS AN ALGEBRAIST Brahmagupta's Corollary also follows at once from the above by putting a'-a, B'-ß and k'=k. N (2aß)+k² (3²+Na²)² Thus the roots are x=2aß and y=²+Na² which is the Corollary. 250 It would be seen that modern historians of mathematics are incorrect when they say that Fermat (1657) was the first to state that the equation Nx+1=y", where N is a non-square integer has an unlimited number of solutions in integers. For this assertion, history takes us to the early Seventh Century A.D. when Brahmagupta wrote his classical treatise, the Brahmasphu- tasiddhanta, and gave the well known two Lemmas and the Corollary to the first Lemma. Second Lemma of Brahmagupta In the Brahmasphuta siddhanta, we find another important Lemma by Brahmagupta stated as follows: On dividing the two roots (of a square Nature) by the square-root of its additive or subtracrive, the roots for interpolator unity (will be found).² This Lemma when expressed in the modern language of algebra would mean that if x-y-ß be a solution of the equation. Nx³+k³=y² then xa/k, yß/k is a solution of the equation Nx¹+1=². This rule, at another place, has been re-enunciated as follows: If the interpolator is that divided by a square then the roots will be those multiplied by its square- Toot, 1. प्रक्ष पशोधक हृते मूले प्रक्ष के रूपे । 2. वर्गच्छिन्ने क्षमे तत्पद्गुणिते तदा मूले । -BrSpSi. XVIII. 65 -BrSpSi. XVIII. 70
