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

240 BRAHMAGUPTA AS AN ALGEBRAIST If the multiplier be negative, it must be made positive; and the additive must be made negative : and then the method of the pulveriser should be employed. Pṛthūdaka Svāmī, however, does not indicate how to derive the solution of the equation. by = -ax+c ...(1) from that of the equation by = ax-c ...(2) The method, however, seems to have been this : Let x=α, y=β be the minimum solution of (2). Then we get bβ = a α-c or b(a-β) = -a(a-b)+c Hence x=a-b, y=a-β is the minimum solution of (1). This rule is very clearly indicated by Bhāskara II and others. We shall give two examples from Bhāskara II (Bījagaṇita) to illustrate the rule : Example I. 13y = -60x + 3 By the method described before, we find that the minimum solution of 13y = 60x+3 is x=11, y=51. Subtracting these values from their respective abraders, namely 13 and 60, we get 2 and 9. Then by the maxim : "In the case of the dividend and divisor being of differ- ent signs, the results from the operation of division should be known to be so", making the quotient negative we get the solu- tion of 13y = -60x+3 as x=2, y=-9. Subtracting these values again from their respec- tive abraders (13, 60), we get the solution of 13y = -60x-3 as x=11, y=-51. Example II. 11y = 18x+10