Converging Series for the Circle , 483. 
either equal to the fum of the two arcs of which { and | • 
are the tangents, or to the difference between the arc of 
which the tangent is f, and the double of the arc of which 
the tangent is -i; that is, putting A=the arc of 45 °. 
A~ 
r 1 
+ — x 
2 
- — 1 Sec. 
+ — X 
L 3 
3-4 5-4“ 7-4 9-4 
1 ~ 3V + 5~<f ~ T 9 1 + 9 T 4 ^ 
Or, A — < 
, 1 1 x 
+ I h ~ 5 : 
3.4 5 4 2 7.4 
1 1 
— x : 1- 
9-4 
1 
See* 
Sec* 
7 3 49 5-49' 7^49 9-49 + 
And the former of thefe values of a is the fame with 
that before mentioned as given by M. euler; but the 
latter is much better, as the powers of ~ converge much 
falter than thofe of f 
corol. From double the former of thefe values of a 
fub trailing the latter, the remainder is, 
A — 
+ 3 X ’ 1 3-9 + 5-9* 7-9* + 9-9 + & ' C ‘ 
1 1 
+ - x : x — 1* 
7 3-49 
7-9* 
1 
+ 
n. ~ Sec. 
5.49 749 9.49’ 
which is a much better theorem than either of the 
former. 
2 . If t be taken =4, then the expreffion gives, 
3 + ’ 
a — 
Here the value of a~\ gives the fame expreffion for the 
value of a as the firft in the foregoing cafe,- and the value 
Vol. LX VI. Sff of 
