* 
Method of correfpondent Values , &c. 
183 
but*- — + 
a- 
z 4 
— &c., and 1 — a — -j — — &c. are 
2 • 3 2 • 3 • 4 • 5 1 . 22 . 3*4 
the fine (j) and cofine (c) of an arc ,2 of a circle whofe radius 
is 1 ; and, confequently, if the fine s and cofine c of an arc a 
be given, the fine of an arc (# + £) = s x (1 — — 4 -— — &c.) 4 -! 
2 2 4 
c(b — 
P 
+ 
&c .), which feries, if b be very fmall 
2 • 3 2 • 3 • 4 • 5 
in proportion to a , converges much fafter than the common 
feries for finding the fine from the arc : it has been given from 
different principles in the Meditationes, and is aifo eafily dedu- 
cible from the feries for finding the fine and cofine from the 
arc by the propofiiions ufually given in plane trigonometry : 
— &c.) 
o $4 
the cofine of the fame arc 0 + J)=:CX(l-— — 
^ ® -4 2 c • 3 • 
^ x C* -r-^+rmri - &c -> 
Ex. 2 . Let the feries be a + b+ a ~r + — — - — • + £cc. 
2 * 3 • 4 • 5 
o o 
2 * D 
f P cfi 0 / b z P o \ 1 
[a 4 f* — + &c. x ( 1 -f — 4- — 4- &c.) 4 - j 
2 * 3 • 4 • 5 v 1 . 2 1 . 2 . 3 . 4 J l 
,4 
2 • 3 
a 
^ + ih + rfri+ &c 0x(4^+-^ + &c.)i 
3 • 4 
3 • 4 
3*4*5 
but a 4 4 - &c.) = x 9 and 1 4 - 4- - — ■ 4- &c. — 
1*2.3 J 1.2 1. 2. 3. 4 
y 
\/ 1 + x l 9 if a be the hyperbolic log, of # 4-V^ 1 + ^ » therefore 
a-^r-b 
a + b-\- — p &c. = x x ( 1 4- ~~ + ;-g— + & c 0 
- • o 2 • 3 • 4*5 v 2 2 * 3 • 4 
I + x X (^4- 4- &C' 
2 • 3 
Z4 
Let ^ 4 - —. 4 - 'f. 4 - &c s r: y, and (# 4 -\/ 14 ^ X 
2 • 3 • 4 * 5 
2 * 3 
(y 4- v/ 1 +/ ) = V, then will a-{-b 4- LlL 
a~\-b' a-^rt) 
2 . 3 2 • 3 • 4 • s 
j” SCCm 
iV-U* 
F f '3 
5. Let 
