finding the Value of an infinite, Series , <kc. 195 
much lefs than the radius; but when the taigent is 
nearly equal to the radius, it converges exceeding {lowly ; 
and when it is quite equal to the radius, or the arch is 
equal to 45 0 , the decreafe of the terms is fo, flow as to 
make the computation of it in the common way, by com- 
puting the value of its initial terms, abfolutely impradi- 
cable. For Sir Isaac newton has obferved concerning 
this feries in that extreme cafe (which then becomes 
equal tor — + — — H + + See.) and another 
u 3 5 7 9 11 13 i 5 J 
feries that is almolf as flow as this, that to exhibit its 
value exact to twenty decimal places of figures, there 
would be occaflon for no lefs than five thoufand mil- 
lions of its terms, to compute which would take up 
above a thoufand years. See Sir Isaac newton’s fecond 
letter to Mr. oldenburgh, dated October 24, 1676, 
in the Commercium Epifiolicum , p. 159. In tliefe 
cafes therefore it will be convenient to make ufe of 
fome artifice to difeover the value of the leries 
f f 
— H 
t t 
73 + 
- + 8:c. ; and we 
t_ r 
3 r F 5 r* 7 9 r u 1 1 137"“ 15 
fliall find the application of the differential feries above- 
mentioned to be a very proper artifice for this purpofe. 
Art. 8. In order to make this application, we muft con- 
,3 y S f Z r 3 f l $ 
fider the feries t- — '4— ■ + + ^ c * 
3 r r 5 7 r 9 r ll r I 3 r 1 3 r 
as being the produd of the multiplication of t into the 
C c 2 feries 
