Did Bhaskar II discover calculus?

Bhaskar II was born in Vijapur in the province of Karnataka in 1114 A.D. He wrote Siddhanta-Shiromani in 1150, which became a classical text in Mathematics and Astronomy. The book is divided in four parts: Lilavati deals with arithmetic, Bijaganita with algebra, Ganitadhyaya and Goladhyaya with astronomy.

In Siddhanta Shiromani, Bhaskar II defines two kinds of planetary velocities: Sthula gati (average speed) and Sukshma or Tatkaliki gati (instantaneous velocity). The process of finding instantaneous velocity involves the use of differential calculus. There is definite proof that Bhaskar II carried out such calculations using the method of differentiation.

According to Hindu astronomy,

l = lmean ± r sina/R


l = true longitude

lmean = mean longitude

r = radius of the epicycle

a = anomaly


R = radius of the deferent cycle

Bhaskar II formulates the expression for the tatkaliki gati (instantaneous velocity) as follows:

"To find the instantaneous velocity (in longitude) of the planet, the kotiphala is to be multiplied by the time rate of change of anomaly and divided by the radius, and the quotient (thus obtained) is to be added to or subtracted from the velocity of the mean planet according as its position is in the six signs from the beginning of Cancer or Capricorn."

Expressed mathematically,

dl/dt = dlmean/dt ± (r cosa/R) da/dt


r cosa = kotiphala

This equation not only provides his familiarity with the notion of differentiation, but also shows his knowledge of the expression

d(sina)/da = cosa

After Bhaskar II, India went through a long hostile foreign rule, and could not produce any mathematician of his caliber for a long time to come.

Reference: D. M. Bose, S.N. Sen and B. V. Subbarayappa, "A Concise History of Science in India", Indian National Science Academy, 1971, p. 203