[ 40i ] 
In order to render this rule fit for ufe in all cafes 
it is only neceflary to know how to find within fuffi- 
cient nearnefs the value of E a? b* and alfo of the 
feries mz — &c*. With refpedt to the former 
Mr. Bayes has proved that, fuppofing K to fignify the 
ratio of the quadrantal arc to it’s radius, E a? bi will 
be equal to 
v/ n 
2 a/K/> 
by the ratio whofe hyperbo- 
lic logarithm is — v - — - 
° 12 n p 
I 
rr 
+ 7 
260 
n 
1 
<1 
I 
1680 
I I 1 
— X — — 77 
360 n p 
~i ~ 
^ n 1 p 1 
~T + 7700 X -4 — ~ — 4" & c * where the nume- 
q I loo n p J q y 
ral coefficients may be found in the following man- 
ner. Call them A, B, C, D, E, &c. Then A — 
2. 2. 3 
10 B + A 
3 4* ^ 2. 4. 5 
2. 6. 7 
.D — 
35 C +21 B + A £ I 
5 2. 8. 9 
1 26 C + 84 D 4- 36 B 4- A 
7 
F 
2. 10 .11 
2. 12. 13 
* A very few terms of this feries will generally give the hyper- 
bolic logarithm to a fufficient degree of exadtnefs. A fimilar fe- 
ries has been given by Mr. 2 De Moivre, Mr. Simpfon and other 
eminent mathematicians in an exprefiion for the fum of the lo- 
garithms of the numbers 1, 2, 3, 4, 5 to a-, which fum they 
have afferted to be equal to l log. c + x + i x log. a- — x + 
tt x — tFo* 3 + t-zVct# 5 &c. c denoting the circumference of a 
circle whofe radius is unity. But Mr. Bayes, in a preceding pa- 
per in this volume, has demonftrated that, though this exprefficn 
will very nearly approach to the value of this fum when only a 
proper number of the firft terms is taken, the whole feries' cannot 
exprefs any quantity at all, becaufe, let .*• be what it will, there 
will be always a part of the feries where it will begin to diverge. 
This obfervation, though it does not much affect the ufe of this 
‘ries, feems well worth the noticeof mathematicians. 462 
