The Science of the Shulba: A Study in Early Hindu Geometry c20.s5.1

c20.s5.1

It would be natural to ask if there is in the *Śulba* any specific rule for determining the approximate value of any surd. But the answer is not very reassuring. For we do not find an express statement of any formula for the purpose. At the same time we can unhesitatingly admit that they had the necessary equipments for the approximate evaluation of surds, at least of some. We have pointed out before that in the science of the *Śulba*, it is sometimes necessary to construct a square having a given area. And that is a geometrical method of finding the square-root of a given number. If the given area is represented by a non-square number, we get a method finding the square-root of a non-square number. One formula follows at once from the method given in the *Śulba*, for the transformation of a rectangle into a square, which has been described before. According to this method, a square (having its side equal to the breadth of the given rectangle) is first cut off from the given rectangle; the excess portion is divided into two halves which are then joined to the two sides of that square. Then by adding a small square in the corner, a large square is completed. The square equivalent to the given rectangle will be obtained, it is said, by subtracting the added small square from the completed large square. This subtraction was doubtless made by the Śulba-workers with the help of the theorem of the square of the diagonal. But it can also be made by cutting off two strips from the two sides, say, the east side and the south side of the completed large square.¹ Suppose A to be the area of the given rectangle and a be the side of the square subtracted from it. Then the side of the large completed square will be a + (A-a²)/2a. The area of the small added square will be ((A-a²)/2a)². Then from each side of the large sphere we shall have to cut off a thin strip of the same breadth. If x denote the breadth of the strip, we must have