Mathematics in Ancient and Medieval India c9.67

c9.67

The idea of such summation of an infinite series or the limit of the sum of its n terms is, in fact, first found in the *Yuktibhāṣā*¹ (16th century A. D.), nearly a century earlier than Newton. For calculating the value of π, the author of *Yuktibhāṣā* considered a square circumscribing a quarter of a circle of radius x. The square was divided by its diagonal. The circumference of the circle included between the diagonal and a side of the square passing through the centre of the circle is ⅛th of the whole circumference. The side of the square covering this portion of circumference was again divided into large number of very small equal parts by lines drawn from the centre of the circle. Let Δ x be the length of each small division on the side of the square which is equal to radius and let n. Δ x = x. In the attempt to calculate the value of : π (since π = circumference/diameter), the author of *Yuktibhāṣā* calculated the value of ⅛th of the circumference by summing up the large number of small chords corresponding to each small equal division of the circle by applying the properties of similar triangles. The expression for the sum of the chords as deduced in *Yuktibhāṣā* is given by :