**Bhaskara II Calculus**

Bhaskara II work, the Siddhanta Shirmoni, is an astronomical treatise and contains many theories not found in earlier works. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry , differential calculus and integral calculus that are found in the work are of particular interest.

Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement. Bhaskara also goes deeper into the vanishes at an extreme value of the function, indicating knowledge of the concept of infinitesimals .

• here is evidence of an early form of Rolle’s theorem in his work

o If then for some with

• He gave the result that if then , thereby finding the derivative of sine, although he never developed the notion of derivatives.

o Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.

• In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1⁄33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.

• He was aware that when a variable attains the maximum value, its differential vanishes.

• He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes. He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem one of the most important theorems in analysis, which today is usually derived from Rolle’s theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara’s Lilavati.

Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara’s work and further advanced the development of calculus in India.

to continue ….

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