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

238- BRAHMAGUPTA AS AN ALGEBRAIST cular case only of the more general latter equation. Of course, there is a little justification also for treating it separately, since both the types of equations represent two different physical conditions of the astronomical problems. In the case of by=ax±c, the conditions are such that the value of either y or x, more particu- larly of the latter, has to be found and the rules for solution formulated with that objective. But in the case of the equation by=ax±1, the physical conditions require the values of both y and x. The equation by=ax±1 is usually known by the name sthira-kuṭṭaka, literally meaning the 'constant pulveriser' Pṛthū- daka Svāmī also names it as dṛḍha-kuṭṭaka meaning firm-pulveri- ser. Later on this term dṛḍha-was confined to another sense, equivlent to nicched (having no divisor) or nirapavarta (irreduci- ble). The origin of the name sthira-kuṭṭaka or constant pulveriser has been explained by Pṛthūdaka Svāmī as being due to the fact that the interpolator (±1) is here invariable. For the solution of this equation, we shall quote Bhāskara I's rule and the rule by Brahmagupta. Bhāskara I writes in this connection as follows : The method of the pulveriser is applied also after subtracting unity. The multiplier and quotient are respectively the numbers above and underneath. Multi- plying those quantities by the desired number divide by the reduced divisor and dividend; the residues are in this case known to be the (elapsed) days and (resid- ues of) revolutions respectively¹. The pulveriser ax - c -------- = y ... (1) b may be written as aX - 1 -------- = Y ... (2) b where x=cX and y=cY. If X=α, Y=β is a solution of (2), then x=cα, y=cβ will be a solution of (1). Hence the above rule.

1. रूपमेकमपास्यापि कुट्टाकारः प्रसाध्यते । गुणकारोऽथ लब्धं च राशी स्यातामुपर्यधः ॥ —MBh. I. 45