पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/४२५

(also) of the square. (That same) quotient, if multiplied by six, gives rise to the required measure of the base of the (equilateral) triangle as also of the longish quadrilateral figure. Half (of this) is the measure of the perpendicular-side (in the case of the longish quadrilateral figure).

An example in illustration thereof.

143-145. A king caused to be dropped an excellent carpet on the floor of (his) palace in the inner apartments of his zenana amidst the ladies of his harem. That (carpet) was (in shape) a regular circle. It was held (in hand) by those ladies. The fist fuls of, both their arms made each (of them) acquire 15 (daṇdas out of the total area of the carpet). How many are the ladies, and what is the diameter (of the circle) here ? What are the sides of the square (if that same carpet be square in shape) ? and what the

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magnitude. The stanza states a rule for finding out the measure of the diameter of the circle, or of the sides of the square, or the equilateral triangle or the oblong. If m represents the area of each part and n the length of a part of the total perimeter, the formulas given in the rule are--

${\tfrac {m}{n}}\times 4=$ diameter of the cirole, or side of the square;

and${\tfrac {m}{n}}\times 6=$ side of the equilateral triangle or of the oblong;

and half of ${\tfrac {m}{n}}\times 6=$ the length of the perpendicular-side in the case of the oblong.

The rationale will be clear from the following equation, where x represents the number of parts into which each figure is divided, a is the length of the radius in the case of the circle, or the length of a side in the case of the other figures; and b is the vertioal side of the oblong:

In the case of the Circle ${\tfrac {x\times m}{x\times n}}={\tfrac {\pi a^{2}}{2\pi a}}$;

In the case of the Square ${\tfrac {x\times m}{x\times n}}={\tfrac {a^{2}}{4a}}$;

In the case of the Oblong ${\tfrac {x\times m}{x\times n}}={\tfrac {a\times b}{2(a+b)}}$; here b is taken to be equal to half of a.

It has to be noted that only the approximate value of the area of the equilateral triangle, as given in stanza 7 of this chapter, is adopted here. Otherwise the formula given in the rule will not hold good.

143-145. What is called fistful in this problem is equivalent to four aṅgulas in measure.