274 NEW SERIES for the 



Thus we have the circular arch, or rather its reciprocal (from 

 which the arch itfelf is eafily found), exprefied by a feries of a 

 very fimple form ; and this is the firfl 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, fee A : I : : tan A 

 — tan 4- A : tan ± A ; therefore, I -f- fee A : I : : tan A : tan ± A, 



tan A 

 and hence tan ± A = —j~r — X * "^ ut as ** ec ^ * s S reater tnan 



i, therefore i -f- fee A mufl be greater than 2, and confequent- 



ly — r~r — r ^ s tnan > hence it follows, that tan 4- A 



J i -j- fee A 2 ' * 



mufl 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 i a, £ a, f a, &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-efficients, it appears, 

 that each term of the feries, after the fecond, is lefs than one- 

 fourth of the term before it j and this is one limit to the rate 

 of convergency of the feries. 



9. Again, to find another limit, let us refume the formula 



tanfA= t , an r A A , from which it follows, that tan ~% A 

 2 1 -f- fee A tan A 



= — r- r — r> and fimilarly, that *-r- == — rr — r-r- But 



1 + fee A J tan 7 A 1 -f- fee \ A 



fince 



