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

224 BRAHMAGUPTA AS AN ALGEBRAIST Hence ' by—ax = R₁—R₂ Putting c = R₁ ∽ R₂ we get by—ax = ± c the upper or lower sign being taken according as R₁ is grea- ter than or less than R₂. Class II : To find a number (x) such that its product with a given number (a) being increased or decreased by another given number (δ) and then divided by a third given number (β) will leave no remainder. This means that in other words, we shall have to get the solution of : ax ± r —————— = y β in positive integers. Class III : Here we have to deal with an equation of the form : by ÷ ax = ± c Kuṭṭaka, Kuṭṭākāra and Kuṭṭa : These are the three terms which Brahmagupta has used in regards to the subject of indeter- minate analysis of the first degree. Āryabhaṭa I has also descri- bed this method in brief, but he does not use the word kuṭṭaka. In the Mahābhāskarīya of Bhāskara I we have the terms kuṭṭā- kāra and kuṭṭa (522 A.D.) MBh. I. 41,49). These words have been translated into English as pulveriser or grinder. According to Datta and Singh, the Hindu method of solving the equation by-ax= ± c is essentially based on a process of deriving from it successively other similar equations in which the values of the coefficients (a,b) become smaller and smaller. Thus the process is indeed the same as that of breaking a whole thing into smaller pieces, and this accounts for its name kuṭṭaka or ‘pulveriser’. In the problems of the Class I, the quantities (a and b) are called’ divisors’ bhāgahāra, bhajaka, cheda etc.) and R₁ and R₂ as ‘remainders’ (agra or śeṣa etc.). while in a problem of the Class II ; β is ordinarily called the ‘divisor’ (bhāgahāra or bhā- jaka) and γ the ‘interpolator’ kṣepa, kṣepaka etc.) ; here a is called the ‘dividend’ (bhājya), the unkown quantity to be found (x) is called the ‘multiplier’ or (guṇaka or guṇakāra etc) and y the