Mr. hutton on Converging Series , Sec. 477 
force of fired gun-powder; as they feem to lead 
to many important conclufions,I believe I fhall,in a 
fhort time, have the honour of fubmitting to your 
infpe&ion an account of fome of them, for the like 
purpofe as the following paper. 
d HE excellency of this method is primarily owing 
to the fimplicity of the feries by which an arc is found 
from its tangent. For if t denote the tangent of an arc a> 
the radius being 1, then it is well known, that the arc cl 
will be equal to the infinite feries, 
t r t s t 7 t 9 t xl 
T _ 7 + s _ 7 + 7~ n +&c - 
where the form is as fimple as can be defired. And it is 
evident, that nothing farther is required than to contrive 
matters fo, as that the value of the quantity t in this feries 
may be both a fmall and a very fimple number. Small, 
that the feries may be made to converge fufficiently faff ; 
and fimple, that the feveral powers of t may be railed 
by eafy multiplications, or eafy divilions. 
Since the firfl difeovery of the above feries, many 
have ufed it, and that after different methods, for de- 
termining the length of the circumference to a great 
number of figures. Among thefe were Dr. halley, Mr. 
abraham sharp, Mr. mach! n, and others, of our own 
country; and M. de lagny, M. euler, See. abroad. Dr. 
HALLEY ufed the arc of 30°, or ~th of the circum- 
ference, the tangent of which being= vd, by fubftituting vd 
for t in the above feries, and multiplying by 6, the femi 
Rrr 2 circum- 
