पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४०४

208

GAṆITASĀRASAṄGRAHA.

An example in illustration thereof.

87${\tfrac {1}{2}}$. In the case of a (regular) six-sided figure each side is 2 daṇdas in measure. In relation to it, what are the squares of the measures of the diagonal, of the perpendicular and of the minutely accurate area of the figure ?

The rule for arriving at the numerical measure of the sum of a number of square root quantities as well as of the remainder left after subtracting a number of square root quantities one from another in the natural order :-

88${\tfrac {1}{2}}$. (The square root quantities are all) divided by (such) a (common) factor (as will give rise to quotients which are square quantities). The square roots (of the square quantities so obtained) are added together, or they are subtracted (one from another in the natural order). The sun and remainder (so obtained) are (both) squared and (then) multiplied (separately) by the divisor factor (originally used). The square roots (of these resulting products) give rise to the sum and the (ultimate) difference of the quantities (given in the problem). Know this to be the process of calculation in regard to (all kinds of) square root quantities.

An example in illustration thereof.

89${\tfrac {1}{2}}$. O my friend who know the result of calculations, tell me the sum of the square roots of the quantities consisting of 16, 36 and 100; and then (tell me) also the (ultimate) remainder in relation to the square roots (of the same quantities). Thus ends the minutely accurate calculation (of the measure of areas).

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88${\tfrac {1}{2}}$. The word karaṇi occurring here denotes any quantity the square root of which is to be found out, the root itself being rational or irrational as the case may be. The rule will be clear from the following working of the problem given in stanza 89${\tfrac {1}{2}}$:-

To find the value of ${\sqrt {16}}+{\sqrt {36}}+{\sqrt {100}}{\text{, and }}{\sqrt {100}}-({\sqrt {36}}-{\sqrt {16}}).$

${\begin{aligned}{\text{These are to be represented as}}&{\sqrt {4}}({\sqrt {4}}+{\sqrt {9}}+{\sqrt {25}});&{\sqrt {4}}&\left\{{\sqrt {25}}-({\sqrt {9}}-{\sqrt {4}}\right\}.\\&={\sqrt {4}}(2+3+5);&=&{\sqrt {4}}\left\{5-(3-2)\right\}.\\&={\sqrt {4}}(10);&=&{\sqrt {4}}(4).\\&={\sqrt {4}}\times {\sqrt {100}};&=&{\sqrt {4}}\times {\sqrt {16}}.\\&={\sqrt {400}};&=&{\sqrt {64}}.\\&=20;&=&8.\end{aligned}}$