# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३९५

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एतत् पृष्ठम् परिष्कृतम् अस्ति
199
CHAPTER VII-MEASUREMENT OF AREAS.

cular (to the base) as also of the segments (of the baso 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.[1] 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 sides. The (resulting) two sums constitute the multiplier 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,O friend who know the secret of calculation, the value of the diagonal and also the accurate value of the area.

54.^  Algebraically represented bhe measure of the diagonal of a quadrilateral figure as given here is:–

${\displaystyle {\sqrt {\tfrac {(ac+bd)(ab+cd)}{ad+bc}}}{\text{ or }}{\sqrt {\tfrac {(ac+bd)(ad+bc)}{ab+cd}}}}$

These formulas also are correct only for cyclic quadrilaterals. Bhāskarācārya 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 |he diagonals. Vide the following stanza from his Līlāvatī:-

लम्बयोः कर्णयोर्वेकमनिर्दिश्यापरान् कथम्।
पृच्छत्यनियतत्वेऽपि नियतं चापि तत्फलम् ।।
स पृच्छकः पिशाचो वा वक्ता वा नितरां ततः ।
यो न वेत्ति चतुबाहुक्षेत्रस्यानियतां स्थितिम् ।