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

BRAHMAGUPTA'S RULES OF ANALYSIS 237.
Brahmagupta further observes :
Such is the process when the quotients (of mutual
division) are even in number. But if they be odd,
what has been stated before as negative should be
made as positive, or as positive should be made
negative.¹
Regarding the direction for dividing the divisor corres-
ponding to the greater number by the divisor corresponding to
the smaller remainder, Pṛthūdaka Svāmī (860A.D.) observes that
it is not absolute, rather optional; so that the process may be
conducted in the same way by starting with the division of the
divisor corresponding to the smaller remainder by the divisor
corresponding to the greater remainder. But in this case of in-
version of the process, he continues, the difference of the remain-
ders, must be negative.
That is to say, the equation
by=ax+c
can be solved by transforming it first to the form
ax=by-c
so that we shall have to start with the division of b by a.
For the details of the "Theory of the pulveriser" as applied
to the problems in Astronomy, the reader is referred to the writ-
ings of Bhaṭṭa Govind, translated by K.S. Shukla, and given as
an Appendix to the edition of the Laghu-Bhāskarīya. For the
rationale of the rules in relation to kuṭṭaka or the pulveriser
operation, one may also refer to the chapters by Datta and Singh
in the History of Hindu Mathematics: Algebra.
Solution of by=ax ± 1.
This simple indeterminate equation has a special use in
astronomical calculations and therefore, Indian algebraists have
paid special attention to it. In fact, this equation is solved
exactly in the same way as the equation by=ax ± c; it is a parti-

1. एवं समेषु विषमेष्वृणं धनं धनमृणं यदुक्तं तत् ।
   ऋणधनयोर्व्यस्तत्वं गुण्य प्रक्षेपयोः कार्यम् ॥ —BrSpSi. XVIII. 13.