पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/३९७

CHAPTER VII-MEASUREMENT OF AREAS. 199 cular (to the base) as also of the segments (of the base caused thereby)? 53. In the case of a scalene triangle one of the sides is 13 (in measure), the opposite side is 15, and the base is 14. What indeed is the calculated measure (of the area of this figure), and what of the perpendicular (to the base) and of the basal segments ? Hereafter (we give) the rule for arriving at the value of the diagonal of the five varieties of quadrilateral figures. 54. The two quantities obtained by multiplying the basal side by the (larger and the smaller of the right and the left) sides are (respectively) combined with the two (other) quantities obtained by multiplying the top side by the (smaller and the larger of the right and the left) des. The (resulting) two sums constitute the multi- plier and the divisor as also the divisor and the multiplier in relation to the sum of the products of the opposite sides. The square roots (of the quantities so obtained) give the required measures of the diagonals. Examples in illustration thereof. 55. In the case of an equilateral quadrilateral which has all around a side measure of 5, tell me quickly, Q friend who know the secret of calculation, the value of the diagonal and also the accurate value of the area. 54. Algebraically represented the measure of the diagonal of a quadrilateral figure as given here 18- /(ac + bd) (ad + bc) - (ac+bd) (ab+cd) ad + bc or N ab + cd These formulas also are correct only for cyclic quadrilaterals. Bhaskarā- cārya is aware of the futility of attempting to give the measure of the area of a quadrilateral without previously knowing the values of the perpendicular or of the diagonals Vide the following stanza from his Lila vati - लम्बयोः कर्णयोर्वैकमनिर्दिश्यापरान् कथम् । पृच्छत्यनियतत्वेऽपि नियतं चापि तत्फलम् !! स पृच्छकः पिशाचो वा वक्ता वा नितरां ततः । यो न वेत्ति चतुर्बाहुक्षेत्रस्यानियतां स्थितिम् ||