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

246 BRAHMAGUPTA AS AN ALGEBRAIST x is known as the lesser root, y is the greater root, N is the multi- plier (guṇaka) and c is interpolator or kṣepaka. Bhāskara II has used the word "hrasvamūla" for kaniṣṭha pada or ādya-mūla lite- rally meaning "lesser root". The earlier terms, the "first root" (ādyamūla) for the value of x and the "second root" or the "last root" antya-mūla for the value of y are quite free from ambiguity Their use is found in the algebra of Brahmagupta. The later terms appears in the works of his commentator Pṛthūdaka Svāmī. Brahmagupta uses the term kṣepa, prakṣepa or prakṣepaka in the sense of "interpolator." Again, when negative, the inter- polator is sometimes distinguished as the "subtractive" or śodhaka and the positive interpolator is then called "the addi- tive." Lemmas of Brahmagupta Prior to our giving the general solution of the Square-nature or Varga-Prakṛti, it would be better to give two Lemmas esta- blished by Brahmagupta. We have the following in the Brāhma- sphuṭasiddhānta: Of the square of the optional number multiplied by the guṇaka and increased or decreased by an other optional number, iṣṭa, (extract) the square root. (Proceed) twice. The product of the first roots multiplied by the guṇaka together with the product of the second roots will give a (fresh) second root; the sum of their cross-pro- ducts will be a (fresh) first root. The (corresponding) interpolator will be equal to the product of the (previ- ous) interpolators.¹ There is a little difficulty in ascertaining the real sense of the rule given in these lines since the word dvidhā (twice) has two implications. Firstly, it may mean that the earlier operations of finding roots are made on two optional numbers with two optio- nal interpolators, and with the results thus obtained the subse-

1. मूलं द्विधेष्टवर्गाद् गुणक गुणादिष्टयुत विहीनाच्च । आद्यवधो गुणकगुणः सहान्त्यघातेन कृतमन्त्यम् ॥ वज्रवधैक्यं प्रथमं प्रक्षेपः क्षेपवध तुल्यः । प्रक्षेपशोधकहृते मूले प्रक्षेपके रूपे ॥ —BrSpSi. XVIII 64-65