# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३८८

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192
GAṆITASĀRASAṄGRAHA.

The rule for arriving separately at the numerical measures of the circumference, of the diameter, and of the area of a circular figure, from the combined sum obtained by adding together the approximate measure of its area, the measure of its circumference and the measure of its diameter:-

30. In relation to the combined sum (of the three quantities) as multiplied by 12, the quantity thrown in so as to be added is 64. of this (second) sum the square root diminished by the square root of the quantity thrown in gives rise to the measure of the circumference.

An example an illustration thereof.

31. The combined sum of the measures of the circumference, of the diameter and of the area (of a circle) is 1116. Tell me what the (measure of the) circumference is, what (that of) the calculated area and what (of) the diameter is.

The rule for arriving at the practically approximate value of surface-areas resembling (the longitudinal sections of) the yava grain, (of) the mardala, (of) the paṇava, and (of) the vajra, the

32. In the case of areas shaped in the form of the yava grain, of the muraja, of the page, and of the vajra, the

30. This rule will be clear from the following algebraical representation:- Let c be the circumference of the circle as ${\displaystyle \pi }$ is taken to be equal to 3, ${\displaystyle {\tfrac {c}{3}}}$is the diameter and ${\displaystyle 3{\tfrac {c^{2}}{36}}}$ is the area of the circle. If m stands for the combined sum of the circumference, the diameter and the area of the circle, then the rule given in the stanza to the effect that ${\displaystyle c={\sqrt {12m+64}}-{\sqrt {64}}}$ may be easily arrived at from the quadratic equation containing the data in the problem:- ${\displaystyle c+{\tfrac {c}{3}}+3{\tfrac {c^{2}}{36}}=m}$

32. Muraja means the same thing as mardala and mṛdaṅga. The shape of the various figures mentioned in this stanza is as follows:

 Yavākārakṣētra. Murajākārak,sētra Paṇavākārakṣētra. Vajrākārakṣētra.