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

BRAHMAGUPTA'S OWN RULE 275 is effected without forming an equation of the factum Why then was it done so ?¹ Datta and Singh think that the reference in the latter por- tion of this rule is to the method of the unknown author : “Kiṁ kṛtaṁ tadataḥ” ? The principle underlying Brahma- gupta's method is to reduce, like the Greek Diophantus (c.275 A.D.), the given indeterminate equation to a simple determi- nate one by assuming arbitrary values for all the unknowns ex- cept one. So undoubtedly it is inferior to the earlier method. We now take an illustrative example from Brahmagupta : On subtracting from the product of signs and degrees of the Sun, three and four times (respectively) those quantities, ninety is obtained. Determining the Sun within a year (one can pass as a proficient) mathema- tician. If we presume x to denote the signs and y the degrees of the Sun, then the equation would be : xy - 3x - 4y = 90 Pṛthūdaka Svāmī solves it in two ways : (i) Let us assume the arbitrary number to be 17. then x = 1/1 ( (90.1 + 3.4) / 17 + 4 ) = 10 y = 1/1 (17 + 3) = 20 (ii) Let us assume arbitrarily y = 20. On substituting this value of y in the above equation, we get 20x - 3x = 170 whence x = 10.

1. भावितके यद्घातो विनष्टवर्णेन तत्प्रमाणानि । कृतेष्टानि तदाहत वर्णैक्यं भवति रूपाणि ॥ वर्ण प्रमाणभावित घातो भर्वातेष्ट वर्ण संख्यैवन् । सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥ —BrSpSi. XVIII. 62-63
2. भानो राश्यंशवधात् त्रिचतुर्गुणितान् विशोध्य राश्यंशान् । नवतिं दृष्ट्वा सूर्यं कुर्वन्नावत्सराद् गणकः ॥ —BrSpSi. XVIII. 61. — : o : — Reference H.T. Colebrooke : Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmagupta and Bhas- cara, London, 1817. B. Datta and A.N. Singh : History of Hindu Mathematics Pt. I and II, 1962.