पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४३१

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एतत् पृष्ठम् परिष्कृतम् अस्ति
235
CHAPTER VII--MEASUREMENT OF AREAS.

the optionally chosen quantity gives rise to the measure of the perpendicular.

An example in illustration thereof.

157${\displaystyle {\tfrac {1}{2}}}$. In the case of an isosceles triangular figure, the accurate measurement of the area is 12. The optionally chosen quantity is 3. Give out quickly, O friend, the values of (its) sides, base and perpendicular.

The for arriving after knowing the exact numerical rule, measure of a (given) area, at a triangular figure with unequal sides, having that same accurately measured area (as its own):-

158${\displaystyle {\tfrac {1}{2}}}$. The given area is multiplied by eight, and to the resulting product the square of the optionally chosen quantity is added. Then the square root (of the sum so resulting is obtained). The cube (of this square roots) is (thereafter) divided by the optionally chosen number and (also) by the square root (obtained as above). Half of the optionally chosen number gives the measure of the base (of the required triangle). The quotient (obtained in the previous operation) is lessened (in value) by the (measure of this) base. (The resulting quantity) is to be used in carrying out the saṅkramaṇa process in relation to the square of the optionally chosen quantity as divided by to as well as the square root (mentioned above). (Thus) the values of the sides are arrived at.

158${\displaystyle {\tfrac {1}{2}}}$. It A represents the area of a triangle, and d is the optionally chosen nunber, then according to the rule the required values are obtained thus :

${\displaystyle {\tfrac {d}{2}}=base;}$
and ${\displaystyle {\tfrac {{\tfrac {({\sqrt {8A+d^{2}}})^{3}}{d{\sqrt {8A+d^{2}}}}}-{\tfrac {d}{2}}\pm {\tfrac {d^{2}}{2{\sqrt {8A+d^{2}}}}}}{2}}=sides;}$

when the area and the base of a triangle are given, the locus of the vertex is a line parallel to the base, and the sides can have any set of values, In order to arrive at a specific set of values for the sides, it is evidently assumed here that the sum of the two sides is equal to the sum of the base and twice the altitude, i.e., equal to ${\displaystyle {\tfrac {d}{2}}+2{\tfrac {A}{d\div 4}}}$. With this assumption, the formula above given for the measure of the sides can be derived from the general formula for the area of the triangle, ${\displaystyle {\sqrt {s(s-a)(s-b)(s-c)}}}$ given in stanza 50 of this chapter.