पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४००

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204

GAṆITASĀRASAṄGRAHA.

root of that (which happens to be the resulting sum here) gives rise to the measure of the (bent) bow-stick. In the case of finding out the measure of the string and the measure of the arrow, a course converse to this is adopted.

The rule relating to the process according to the converse (here mentioned):-

74 ${\tfrac {1}{2}}$ . The measure of the arrow is taken to be the square root of one-sixth of the difference between the square of the string and the square of the (bent stick of the) bow. And the square root of the remainder, after subtracting six times the square of the arrow from the square of the (bent stick of the) bow, gives rise to the measure of the string.

An example in illustration thereof.

75 ${\tfrac {1}{2}}$ . In the case of a figure having the outline of a bow, the string-measure is 12, and the arrow-measure is 6. The measure of the bent stick is not known. You (find it out), O friend. (In the case of the same figure) what will be the string-measure (when the other quantities are known), and what its arrow-measure (when similarly the other requisite quantities are known) ?

The rule for arriving at the minutely accurate result in relation to figures resembling a Mṛudaṅga, and having the outline of a Paṇava, and of a Vajra-

76 ${\tfrac {1}{2}}$ .^[1] To he resulting area, obtained by multiplying the (maximum) length with (the measure of the breadth of the mouth, the value of the areas of its associated bow-shaped figures is added. The resulting sum gives the value of the area of a figure resembling (the longitudinal section of) a Mṛdaṅga. In the case

In giving the rule for the measure of the arc in terms of the chord and the largest perpendicular distance between the arc and the chord, the arc forming a semicircle is taken as the basis, and the formula obtained for it is utilized for arriving at the value of the arc of any segment. The semicircular arc $=r\times {\sqrt {10}}={\sqrt {10r^{2}}}={\sqrt {6r^{2}+4r^{2}}}$ : based on this is the formula for any arc where p = the largest perpendicular distance between the arc and the chord, and c = the chord .

76 ${\tfrac {1}{2}}$ .^ The rationale of the rule here given will be clear from the figures given in the note under stanza 32 above.