Mathematics in Ancient and Medieval India c9.61

c9.61

Although the integral calculus, as we know it to-day arose with Newton and Leibniz in the 17th century A. D., the early stages of its beginning have been traced to the works of Greek scholars namely Antiphon (c. 430 B. C.), Euclid (300 B. C.) and Archimedes (3rd century B. C.)¹. It is in the works of Archimedes that we find the nearest approach to the concept of actual integration. He calculated the area of a parabolic segment by dividing it into a very large number of small plane geometrical figures and summing up their areas when they are infinitely small². The invention of the concept of calculus was therefore attributed to Archimedes. In India, several centuries after Archimedes, Bhāskara II (1150 A. D) in his *Golādhyāya* has given a similar method of finding the area of a circle, surface area of a sphere and volume of the sphere which involve the ideas of integration³. In this method he assumed that the more or less correct result is obtained in these cases by summation of a very large number of small units in the form of triangles for the circle with common vertex at the centre and base on the circumference of the circle and in the form of pyramids with their vertices meeting at the centre and bases lying on the surface for the volume of a sphere. For calculating the surface-area of a sphere, the method suggested was the summation of the measure of the annular spaces formed by drawing large number of circles of increasing radii around any point on the surface of the sphere covering upto the section passing through the centre of the sphere and multiplying the result by 2. The area of each annular space was calculated by the rule *lambaguṇam kumukhayogārdhamiti* (i.e., the rule for finding the area of a trapezium). All these obviously involved the idea of summation of the infinitesimal quantities. Hence the results given are in good agreement with the modern values namely,