518 Mr. Wursu on the Hindi Quadrature of the Circle. 
The lines are as follow: 
Vydsardhénahatadabhishta gunatah Cotyaptamadyam phalam 
Jydvargéna vinighnamadimaphalam tattat phalancha harél 
Crityd coti gunasya tatratuphaléshwécatripanché dibhi 
Bhuctéshwojayutésty yét samayutim jivddhanussishtyaté. 
‘© Multiply the given sine by radius, and divide the product by cosine for the first 
“ quote; multiply this quote by the square of the sine, and divide the product by the 
‘« square of the cosine for the second quote: multiply and divide this last quote, and so 
“ continually each obtained quote, by the square of the sine and the square of the cosine 
“* respectively, and the quotes obtained by this means divide in succession by the odd 
“ numbers 1, 3, 5, 7, 9, 11, &c.; then the sum of the 2d, 4th, 6th, 8th, &c. quotes 
“« being subtracted from the sum of the Ist, 3d, 5th, 7th, 9th, &c. quotes, the remainder 
«© will be the are of the sine which was taken.” 
Note.—If the cosine be less than the sine of the given angle, change the names of 
the two, and proceed as laid down in the rule. 
The infinite series thus beautifully and concisely expressed in the Sanscrit 
sloca, is the following : 
FNS) Wher OT, PEGE BURST UN SE 
7 = Cos. 3.cos? us Bcos®  Tcos.? fe 9cos? eh 
Where a=arc; r=radius; s=sine, and cos=cosine. 
In this series, the first quote equals the tangent of the arc=/ ; the second 
quote equals tangent cubed, divided by thrice radius squared; the third 
equals tangent to the fifth power divided by 5 into radius to the fourth power ; 
or, algebraically, thus: 
te a Bea 
6=t~sateaure ioe O° 
Which is easily proved to be true by a process in fluxions, which demon- 
strates the fact that the fluxion of the tangent of an arc is ¢o the fluxion of the 
arc itself as the square of the secant is to the square of the radius ; in which 
case, the fluxion of the arc is proved to be equal to - and if 7? be 
r+e 
divided by 72+2, the quotient will be ¢ ae ; and 
of r rT 
2 4 £ ad A e 
the fluents of each being taken, it becomes t —s—>+En-7qpitgp- & = 4 = 
the arc itself, as is laid down in the Carana Padhati. 
