274 NEW SERIES for the 
Thus we have the circular arch, or rather its reciprocal (from 
which the arch itfelf is eafily found), exprefled by a feries of a 
very fimple form ; and this is the firft formula which I propo- 
fed to give for the rectification of the circle. 
8. WE now proceed to inquire what is the degree of conver- 
gency of this feries. In the firft place, it appears, that the nu- 
meral co-efficients of the terms are each one-half of that which 
goes before it. Again, A being any arch of a circle, we have 
by a theorem in the elements of geometry, fecA:1::tanA 
—tantA:tantA; therefore, 1-+fecA:1::tanA:taniA, 
tac. faa ? 
and hence tantiA= glee ee But as fec A is greater than 
1, therefore 1 + fec A muft be greater than 2, and confequent- 
tan A tan A 
rarer ra lefs than 
muft be lefs than tan A. Thus it appears, that a being any 
arch lefs than a quadrant, the tangent of any one of the feries 
of arches + 4,44, 74, &c. is lefs than half the tangent of the 
arch before it. By combining the rate of convergency of the 
tangents with that of their numeral co-efhicients, it appears, 
that each term of the feries, after the fecond, is lefs than one- 
fourth of the term before it ; and this is one limit to the rate 
of convergency of the feries. 
3 hence it follows, that tant A 
g. AGAIN, to find another limit, let us refume the formula 
=f ype tee CATIA CARE: tans A 
ee ee teen from which it follows, sa Te 
ine tan Ae. I 
, and fimilarly, that fa TA Pees But 
I 
™~ r+fecA 
fince 
