BRAHMAGUPTA AS AN ALGEBRAIST We can compose this solution with the previous ones, and get another solution, and thus proceed on to innumerable solu- tions. From Brahmagupta's Corollary to First Lemma we get another set of solutions. If (a, b) be solution of the Square- nature, then another solution of it is x=2ab, and y=b+Na² 248 Thus even if we have only one solution, we can get the other solution also (since N is known), and thus we can get any number of solutions one after the other by this Principle of Composition. Brahmagupta's Lemmas have been described by Bhaskara II (1150 A.D.) in the following words: Set down successively the lesser root (hrasva), greater root (jyeştha) and interpolator (kṣepaka); and below them should be set down in order the same or an another (set of similar quantities). From them by the Principle of Composition (Bhavana) can be obtained numerous roots. Therefore the Principle of Composi- tion will be explained here. (Find). the two cross-pro- ducts (vajrabhyasa) of the two lesser and the two greater roots; their sum is a lesser root. Add the product of the two lesser roots multiplied by the prakrti to the product of the two greater roots. the sum will be a greater root. In that (equation) the interpolator will be the product of the two previous interpolators. Again the difference of the two cross-products is a lesser root. Subtract the product of the two lesser roots multiplied by the prakyti from the product of the two greater roots; (the difference) will be greater root. Here also the interpolator is the product of the two (previous) interpolators.¹ 1. हृस्वज्येष्ठ न पकान् व्यस्य तेषां तानन्यान् वाsधो निवेश्य क्रमेण । साध्यान्येभ्यो भावनाभिर्बहूनि मूलान्येषां भावना प्रोच्यतेऽतः ॥ वज्राभ्यासौ ज्येष्ठलन्चोस्तदेक्यं हृस्वं लघ्वोराहतिश्च प्रकृत्या | क्षुण्या ज्येष्ठाभ्यास युग् ज्येष्ठमूलं तत्राभ्यासः क्षेपयोः च पकः स्यात् ।। हवं वज्राभ्यासयोरन्तरं का लोर्षातो यः प्रकृत्या विनिम्नः । मातो यश्च ज्येष्ठयोस्तद्वियोग ज्येष्ठं पोत्रापिच क्षपघातः || Bhaskara II, Brjaganita. VargaPrakṛti. 2-4
