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CHAPTER VII--MEASUREMENT OF AREAS.

Subject of treatment known as the Janya operation.

Hereafter we shall give out the janya operation in calculations relating to measurement of areas. The rule for arriving at longish quadrilateral figure with optionally chosen numbers as bījas:-

90 ${\tfrac {1}{2}}$ .^[1] In the case of the optionally derived longish quadrilateral figure the difference between the squares (of the bīja numbers) constitutes the measure of the perpendicular-side, the product (of the bīja numbers) multiplied by two becomes the (other) side, and the sum of the squares (of the bīja numbers) becomes the hypotenuse.

Examples in illustration thereof.

91 ${\tfrac {1}{2}}$ . In relation to the geometrical figure to be derived optionally, 1 and 2 are the bījas to be noted down. Tell (me) quickly after calculation the measurements of the perpendicular-side, the other side and the hypotenuse.

92 ${\tfrac {1}{2}}$ . Having noted down, O friend, 2 and 3 as the bījas in relation to a figure to be optionally derived, give out quickly, after calculating, the measurements of the perpendicular-side, the other side and the hypotenuse.

Again another rule for constructing a longish quadrilateral figure with the aid of numbers denoted by the name of bījas:-

93 ${\tfrac {1}{2}}$ . The product of the sum and the difference of the bījas forms the measure of the perpendicular-side. The saṅkramaṇa of

90 ${\tfrac {1}{2}}$ .^ Janya literally means “arising from” or “apt to be derived” ; hence it refers here to trilateral and quadrilateral figures that may be derived out of certain given data. The operation known as janya relates to the finding out of the length of the sides of trilateral and quadrilateral figures to be so derived.

Bīja, as given here, generally happens to be a positive integer. Two such are invariably given for the derivation of trilateral and quadrilateral figures dependent on them.

The rationale of the rule will be clear from the following algebraical representation :-

If a and b are the bīja numbers, then $a^{2}-b^{2}$ is the measure of the perpendicular, 2 ab that of the other side, and $a^{2}+b^{2}$ that of the hypotenuse, of an oblong. From this it is evident that the bījas are numbers with the aid of the product and the squares whereof, as forming the measures of the sides, a right-angled triangle may be constructed.