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

BRAHMAGUPTA'S RULES OF ANALYSIS 239 Brahmagupta's Rule in this connection is as follows : Solution of by=ax-1 : Divide them (i.e., the abraded coefficient of the multi- plier and the divisor) mutually and set down the quo- tients one below the other. The last residue (or the reciprocal division after an even number of quotients has been obtained) is multiplied by an optional integer such that the product being diminished by unity will be exactly divisible (by the divisor corresponding to that residue). The (optional) multiplier and then this quotient should be set down (underneath the listed quotients). Now proceeding from the lower most term to the uppermost, by the penultimate multiply the term just above it and then add the lowermost number. (The uppermost number thus calculated being divided by the reduced divisor, the residue (is the quan- tity required. This is the method of the constant pul- veriser¹. Solution of by+ax=±c Indian algebraists usually transformed this equation as by=-ax+c, so that it appeared as a particular case of by=ax+c, in which a was negative. Brahmagupta has been the first person to solve this equation, but the rule given by him is obscure : The reversal of the negative and positive should be made of the multiplier and interpolator.² Pṛthudaka Svāmī has tried to explain it, but he too is not very clear. He says :

1. हृतयोः परस्परं यच्छ्रेषं गुणकार भागहारकयोः । तेन हृतौ निश्छेदौ तावेव परस्परं हृतयोः ॥ लब्धमधोऽधः स्थाप्यं तथेष्ट गुणकारसङ्गुणं शेषम् । शुद्धस्यति यथैकहीनं गुणकः स्थाप्यः फलं चान्त्यम् ॥ अश्रान्तमुपान्त्येन स्वोर्ध्वो गुणितोऽन्त्य संयुतो भक्तम् । निःशेषभागहारेणैव स्थिरकुट्टकः शेषम् ॥ —BrSpSi. XVIII. 9-11
2. एवं समेषु विषमेष्वृणं धनं धनमृणं यदुक्तं तत् । ऋणधनयोर्व्यस्तत्वं गुण्यप्रक्षेपयोः कार्यम् ॥ BrSpSi XVIII. 13