Mathematics in Ancient and Medieval India c6.s5.30

c6.s5.30

Datta has, however, explained that the value of √2 was possibly obtained from a geometrical method.³ This method consists in constructing a square with area equal to the sum of the areas of the two other squares having sides of one unit in length. For this, one of the two squares having side of unit length is divided into three equal parts by lines drawn parallel to one of its side. Each of these parts forms a rectangular piece of one unit in length and one third unit in width. Two of these rectangular parts are then joined lengthwise to the two adjacent sides of the other unit square. This leaves a square hole at one of the corners of the enlarged unit square. This square·hole will have a side of one-third unit in length. The remaining rectangular piece of the divided unit square is again subdivided into three equal parts each forming a square of side one-third unit length. One of these square is fitted into the square-hole formed at one of the corners of the enlarged squāre. Each of the remaining two squares is again subdivided into four equal rectangular pieces having length of ⅓ unit and width of 1/3.4 unit. Eight of these small rectangular slips are placed lengthwise side by side on the two adjacent sides of the enlarged square with four on each side. This again leaves a square-hole at the corner having a side of length 1/3.4 unit. Now two equal thin slips have to be deducted from the two adjacent sides of the enlarged square under construction, the width of each of the slips is therefore given by $((1/3.4)²)/(2 × (1 + ⅓ + 1/3.4)) = 1/3.4.34$