214 JBhdskan'i s knowledge of the Differential Calculus. [No. 3. 



Bhaskaracharya says that " the difference between the longitudes 

 of a planet found at any time on a certain day and at the same 

 time on the following day is called its rough motion during that 

 interval of time ; and that its Tdtkdlika motion is its exact motion." 

 The Tdtkdlika or instantaneous motion of a planet is the motion 

 which it would have in a day, had its velocity at any given instant 

 of time remained uniform. This is clear from the meaning of the 

 term Tdtkdlika and it is plain enough to those who are acquainted 

 with the principles of the Differential Calculus that this Tdtkdlika 

 motion can be no other than the differential of the longitude of a 

 planet. This Tdtkdlika motion is determined by Bhaskaracharya 

 in the following manner. 



" Suppose, x, x = the mean longitudes of a planet on two succes- 

 sive days ; 

 y t y = the mean anomalies ; 

 u, t/ = the true longitudes and 



a = eccentricity or the sine of the greatest equa- 

 tion of the orbit. 

 Then, x — x = the mean motion of the planet, y — y = the motion 

 of the mean anomaly and u' — u = the true motion of the planet." 

 Now according to Bhaskaracharya, the equation of the orbit on the 



a. sin y 



first day = , and 



Kad 



a sin y 

 that on the next day 



and u' = x'-\- 



Ead 

 a. sin y 



, (1). 



Ead 

 a. sin y 



Ead 



a (sin y' — sin y) 

 .'. u — u = x f — x ±: (2). 



Ead 



Now, in order to know the instantaneous value of v! — u, it is 

 necessary first to know the instantaneous value of the Bhogya- 

 khanda or the difference between two successive sines given in 

 Tables of sines. Thus, suppose the sines of the arcs 0, J, 2J 7 3 A, 

 &c. are given in the Tables of sines, then 



