पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४०१

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CHAPTER VII--MEASUREMENT OF AREAS.

of those two (other) figures which resemble (the longitudinal section of) the Paṇava, and (of) the Vajra, that (same resulting area, which is obtained by multiplying the maximum length with the measure of the breadth of the mouth)is diminished by the measure of the areas of the associated bow-shaped figures. (The remainder gives the required measure of the area concerned.)

Examples in illustration thereof.

77${\tfrac {1}{2}}$. In the case of a figure having the outline configuration of a Mṛdaṅga, the (maximum) length is 24; the breadth of (each of) the two mouths is 8; and the ( maximum) breadth in the middle is 16. What is the area ?

78${\tfrac {1}{2}}$. In the case of a figure having the outline of a Paṇava, the (maximum) length is 24; similarly the measure (of the breadth of either) of the two mouths is 8; and the central breadth is 4. What is the area ?

79${\tfrac {1}{2}}$. In the case of a figure having the outline of a Vajra, the (maximum) length is 24; the measure (of the breadth of either) of the two mouths is 8; and the centre is a point. Give out as before what the area is.

The rule for arriving at the minutely accurate value of the areas of figures resembling (the annulus making up) the rim of a wheel, (resembling) the crescent moon and the (longitudinal) section of the tusk of an elephant:-

80${\tfrac {1}{2}}$. In the case of (a circular annulus resembling ) the rim of a wheel, the sum of the measures of the inner and the outer curves is divided by 6, multiplied by the measure of the breadth

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80${\tfrac {1}{2}}$. The rule here given for the area of an annulus, if expressed algebraically, comes to be ${\tfrac {a_{1}+a_{2}}{6}}\times p\times {\sqrt {10}}$${\text{ where, }}a_{1}{\text{ and }}a_{2}$, are the measures of the two circumferences, and p is the measure of the breadth of the annulus. On a comparison of this value of the area of the annulus with the approximate value of the same as given in stanza 7 above (vide note thereunder), it will be evident that the formula here does not give the accurate value, the value mentioned in the rule in stanza 7 being itself the accurate value. The mistake seems to have arisen from a wrong notion that in the determination of the value of this area, $\pi$ is involved even otherwise than in the values of $a_{1}{\text{ and }}a_{2}$.