156 POWERS or tue SINES, 
the ultimate ratio which the fum of the given power of the 
fines bears to a known power of the radius. 
Having proceeded fo far as the 6th power the law of 
continuation became evident ; fo that, fhould any problem 
in mathematical philofophy require it, we may proceed 
as far as we pleafe in fumming the powers of the fines. The 
law is this, 
Make a fraction whofe denominator is the index of the 
given power, and its numerator the fame index, diminifh- 
ed by unity, and multiplied by the fquare of the radius ; 
by this fra€tion multiply the fum of the next but one 
lower power, and we have the fum of the given power. 
Thus rft, the fum of the 1ft power of the fines } 
is=rr, or the fquare of the radius G ees 
2d, fum of the 2d, power or fquares is { — fration. 
=irrx by the arch of go’. 
3d, fum of the 3d, power or cubes is) 
2rrof, the) if, or = 41" 
4th. fum of 4th. powers is =irr of the 2d 
or=irt X by the arch of go’. s 
sth, fum of sth. powers is=rr of the 3d, | 
ros ghaest ob a 
6th, fum of 6th. powers is= {rr of the 4th | 
or=_.r° x by arch of go®. J 
7th, fum of 7th. powers is=*rr of the 5th, } 
or—= sour b By the Law of 
Sth, fum of 8th. powers is=Zrr of the 6th, | Continuation 
or=2ir° x by the arch of go”. 
&e. &, ‘ 
Should your leifure permit you to give any attention fo 
this fubjeét I fhall be glad to fee you furnifh a demonfira= 
tion for the 3d, or any fubfequent cafe abovementioned. 
lam, Sir, 
Your moft obedient humble fervant, 
DAVID RITTENHOUSE. 
Index. 
{ee 
By Infinite 
Series. 
