ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

248 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=b²+Na² 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 Bhāskara II (1150 A.D.) in the following words : Set down successively the lesser root (hrasva), greater root (jyeṣṭha) 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 (Bhāvanā) can be obtained numerous roots. Therefore the Principle of Composi- tion will be explained here. (Find), the two cross-pro- ducts (vajrābhyāsa) 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 prakṛti 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 prakṛti 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. ह्रस्वज्येष्ठ क्षेपकान् न्यस्य तेषां तानन्यान् वाऽधो निवेश्य क्रमेण । साध्यान्येभ्यो भावनाभिर्बहूनि मूलान्येषां भावना प्रोच्यतेऽतः ।। वज्राभ्यासौ ज्येष्ठलघ्वोस्तदैक्यं ह्रस्वं लघ्वोराहतिश्च प्रकृत्या । क्षुण्णा ज्येष्ठाभ्यास युग् ज्येष्ठमूलं तत्राभ्यासः क्षेपयोः क्षेपकः स्यात् ।। ह्रस्वं वज्राभ्यासयोरन्तरं वा लघ्वोर्घातो यः प्रकृत्या विनिघ्नः । घातो यश्च ज्येष्ठयोस्तद्वियोगो ज्येष्ठं क्षेपोत्रापि च क्षेपघातः ।। Bhāskara II, Bījaganita, VargaPrakṛti, 2-4