216 Blidskara's knowledge of the Differential Calculus. No. 3. 



" After having thus determined the T&tMlika Bhogya-khdnda, the 

 instantaneous value of sin y f — sin y is found by the following pro- 

 portion. 



A. cos y cosy x (y • — y) 



As A : : : y' — y ; (== the instantaneous 



E R 



value of sin y — sin y.) 



By substituting the instantaneous value of sin y' — sin y in the 



equation (2), the instantaneous value of u — u, the true motion of 



the planet will be found : that is, 



a. cos y y' — y 



u' — u = x — x± *. 1 (3) 



R R 



This is the instantaneous motion of the planet." 

 This is the way in which Bhaskaracharya determined the instan- 

 taneous motion of the sun and the moon. 



Equation (3) is just the differential of equation (1). As, 



a. sin y 



d (u) = d (x =fc — ) ; 



R. 



a cos y 



or du = d x ± — . . dy ; 



R R 



which is similar to equation (3). 

 Now, the term Tatlcdlika applied by Bhaskaracharya to the velo- 

 city of a planet, and his method of determining it, correspond exactly 

 to the differential of the longitude of a planet and the way for 

 finding it. Hence it is plain that Bhaskaracharya was fully acquaint- 

 ed with the principle of the Differential Calculus. The subject, 

 however, was only incidentally and briefly treated of by him ; aud 

 his followers, not comprehending it fully, have hitherto neglected 

 it entirely. 



I have the honor to be, 



Your obedient servant, 



Bapu Deva JShastri, 

 Ul May, 1858. 



