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

270 BRAHMAGUPTA AS AN ALGEBRAIST Put in other words, this means that one has to solve the following equations : (i) 5x—25 = y² (ii) 10x—100 = y² (iii) 83x—7635 = y² Pṛthūdaka Svāmī, the commentator on the Brāhmasphuṭa- siddhānta proceeds to solve these equations as follows : (1.1) Suppose y = 10; then x = 125. Or put y = 5; then x = 10. (2.1) Suppose y = 10; then x = 20. (3.1) Assume y = 1; then x = 92. He then remarks that by virtue of the multiplicity of suppositions there will be an infinitude of solutions in every case, But no method has been given either by Brahmagupta or his commentator to obtain the general solution. Double Equations of the First Degree Perhaps we have the earliest reference of the simultaneous indeterminate quadratic equations of the type x ± a = u² x ± b = v² in the Bhakaśālī Manuscript (Folio 59, recto). Brahmagupta gives the solution of such simultaneous inde- terminate quadratic equations of a general case as follows : The difference of the two numbers by the addition or subtration of which another number becomes a sq- uare, is divided by an optional number and then incre- ased or decreased by it. The square of half the result diminished or increased by the greater or smaller (of the given number) is the number (required).¹ Expressed in the language of algebra, shall have : = ½ { ½ ( (a - b)/m ± m ) }² ∓ a

1. याभ्यां कृतिरधिको नस्तदन्तरं हृत युतो न मिष्टेन । तद्दल कृतिरधिकोऽधिकयो रविको न यो राशिः ॥ —BrSpSi. XVIII. 74