278 NEW SERIES for the 
which formula, by reduction, will be found to be the very fame 
as the other. 
12, Tue refult, then, of the whole inveftigation, may be 
briefly ftated as follows: Let @ denote any arch of a circle of 
which the radius is unity, then fhall 
I I I at: I 
Meant a+ ltanta+itanta+ td tanta... 
_ janat Sy rae rok in etna Neagle eae ges 
I 
a 
+ T(m) + Tom+ 1) + SIE 
af Tim) and ‘['”(m+1) denote ‘any two fucceeding terms 
of the feries : tan : a+ i tan : a+, &c., their places in it 
being exprefled by the numbers m and m+1; and where S$ is 
put for the fum of all the remaining terms; and the limits of S 
are the two Ripe 
= 5 Tots) and - 2 Peat — (Toy~4 T(m+1)) T(m+x) 
Sea a ee 
3(T (m) — Tin+1)) y 
is, S is lefs than the former, but greater than the latter. 
The expreflions tan — 4, tan z a, tan 5% &e, are aes de- 
2 4 
ducéd from tan a, and from one ‘another, by a wall known for 
mula in the arithmetic of fines, which may be exprefled ‘thus, 
: te oatoy ides sedis Mp amit: <p 
ope A SN ap avoy tan Ay 
13. I now proceed to the inveftigation of a fecond formula 
for the rectification of the circle ; and for this purpofe refume 
the 
