पृष्ठम्:Ganita Sara Sangraha - Sanskrit.djvu/४०७

CHAPTER VII-MEASUREMENT OF AREAS. 209 Subject of treatment known as the Janya operation. Hereafter we shall give out the junya operation in calculations. relating to measurement of areas. The rule for arriving at a longish quadrilateral figure with optionally chosen numbers as bijas :-- 90. In the case of the optionally derived longish quadrilateral figure the difference between the squares (of the bija numbers) constitutes the measure of the perpendicular-side, the product (of the bija numbers) multiplied by two become the (other) side, and the sum of the squares (of the bija numbers) becomes the hypotenuse. Examples in illustration thereof. 91. In relation to the geometrical figure to be derived option- ally, 1 and 2 are the bas to be noted down. Tell (me) quickly after calculation the measurements of the perpendicular-side, the other side and the hypotenuse 92 Having noted down, O friend, 2 and 3 as the bijas in rela- tion to a figure to be optionally derived, give out quickly, after calculating, the measurenients 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 bējas:- 93. The product of the sum and the difference of the bigas forms the measure of the perpen licular-side The sankramana of 90 Janya literally means "anising from " o "apt to be der ', hence it refers here to tiilateral and quadrilateral figures that may be derived out of certau 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 Bija, as given here, generally happens to be a positive integer. Two such are invariably given for the derivation of tilateral and quadrilateral figures dependent on them. The rationale of the rule will be clear from the following algebraicai representation - If a and b are the bija numhers, then a² ba 1s the measure of the perpendi- cular, 2 ab that of the other side, and a² + b² that of the hypotenuse, of an oblong. From this it is evident that the bigas 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.