1858.] Blidslcartis knowledge of the Differential Calculus. 



215 



sin A — sin 0, sin 2A — sin A, sin SA — sin 2A, &c. are the Bhogya- 

 Miandas. 



" These are not equal to each other but gradually decrease, and 

 consequently while the increase of the arc is uniform, the increment 

 of the sine varies" — on account of the deflection of the arc. Hence 

 the difference between any two successive sines is not the Tdtlcdlilca 

 Bliogya-lclianda ; but if the arc instead of being deflected be increased 

 in the direction of the tangent then the increase which would take 

 place in the sine is the Tdtlcdlilca Blwgya-hlianda i. e. the instan- 

 taneous motion of the sine. 



Thus, in the accompanying 

 diagram, suppose the arc Df 

 = A, then, sin Af— sin AD = 

 fg — DTI = fin, the Bliogya- 

 lclianda of the sine DE ; but 

 this is not the Tdtlcdlilca Blio- 

 gya-lclianda of that sine. If the 

 arc AD instead of being de- 

 flected towards^, be increased 

 in the direction of the tangent, 

 so that DF=Df= A; then 

 jig. — DE=Fn, which would be the Tdtlcdlilca Bliogya-lclianda of 

 the sine DIE i. e. the instantaneous motion of that sine." 



Bhaskaracharya has determined that " the Tdtlcdlilca Bhogya- 

 hhanda varies as the cosine of arc, i. e. when arc = 0, its cosine 

 equals the radius, and A = the Tdtlcdlilca Bliogya-lclianda. And, as 

 the arc increases, the cosine and the Bliogya-klianda decrease. Hence, 

 if y be any given arc, the Tdtlcdlilca Bhogya Ichanda answering to 

 it will be found by the following proportion. 



As, R (or the cosine of an arc = 0.) 



The Tdtlcdlilca Bliogya-lclianda (= A.) 



Cosine y. 



Tdtkdlilca Bliogya-lclianda of sin y. 



A. cos y. 

 Tdtlcdlilca Bhogya-lcJianda = 



JZ 



The reason of the above proportion can be easily understood 

 from the two similar triangles DCE and DFn in the above diagram. 



