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

262 BRAHMAGUPTA AS AN ALGEBRAIST The sum of the squares of two unequal numbers is the side; their product multiplied by two is the altitude, and twice the difference of the squares of those two unequal numbers is the base of an isosceles triangle.¹ Thus if m,n be two integers such that m is not equal to n, the sides of all rational isosceles triangles with integral sides are given by m² + n², m² + n², 2(m² - n²) and the altitude of the triangle is 2mn. This method was also followed by Mahāvīra and other Indian mathematicians. In fact, their solutions are based on the juxtaposition of two rational right triangles, equal so that they have a common leg. It is remarkably a powerful device, for every rational triangle or quadrilateral may be formed by the juxtaposition of two or four rational right triangles. Isosceles Triangles with a Given Altitude Here we have a rule given by Brahmagupta for finding out all rational isosceles triangles possessing the same altitude : The (given) altitude is the producer (karaṇī). Its square divided by an optional number is increased and diminished by that optional number. The smaller is the base and half the greater is the side.² Thus if m be any rational number then for a given definite altitude a, the sides of the rational isosceles triangles are ½ (a²/m + m) each and the base is (a² - m²)/m . We shall illustrate it by an example taken from the commentary of Pṛthūdaka Svāmī The given altitude is 8; let us take any rational number m = 4 then the two equal sides of the isosceles are given by ½ ((8² + 4²)/4) = 10 each and the base is (8² - 4²)/4 = 12. Thus the three sides of the

1. कृति युतिर सदृशराश्योर्बाहुर्घातो द्विसंगुणो लम्बः । कृत्यन्तरमसदृशयोर्द्विगुणं द्विसमत्रिभुज भूमिः ॥ —BrSpSi. XII. 33
2. करणी लम्बस्तत्कृतिरिष्टहतेष्टोन संयुताऽल्पा भूः । अधिको द्विहृतो बाहुः संक्षेप्यो यद्वधो वर्गः ॥ —BrSpSi. XII. 37.