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

264 BRAHMAGUPTA AS AN ALGEBRAIST Thus the two sides of the rational scalene triangle are 15 and 13. The base is . ½ ( 12²/6 - 6 ) + ½ ( 12²/8 - 8 ) = 9 + 5 = 14 The altitude is m = 12; area is equal to (base × altitude) / 2 = (14 × 12) / 2 = and the segments are √(13² - 12²) = 5 and √(15² - 12²) = 9. Thus they are all integers. Rational Isosceles Trapeziums Brahmagupta has given us a method of obtaining such isosceles trapeziums whose sides, diagonals, altitude, segments and area are all rational numbers. His rule is as follows : The diagonals of the rec- tangle (generated) are the flank sides of an isosceles trapezium; the square of its side is divided by an optional number and then lessened by that optional number and divided by two; (the result) increased by the upright is the base and lessened by it is the face.¹ Fig. 21 Here in the figure, we have the isosceles trapezium ABCD of which C D is the base and A B is known as the fase. Accord- ing to Brahmagupta's rule, we have ( p being the optional number). CD = ½ ( (4m²n² - p) / p ) + ( m² - n² ) (base) AB = ½ [ 4m²n²/p - p ] - [ m² - n² ] (face) DH = ( m² - n² ) (upright)

1. आयतकर्णौ बाहू भुजकृतिरिष्टेन भाजितेष्टोना । द्विहृता कोट्यधिका भूर्मुखमूना दिसमचतुरस्रे ॥ —BrSpSi. XII. 36