of the Hyperbolic Logarithm of 10. 
5 39 
f— 
4 o 
40 
5.20 * 
9.20 
13.20 
+ ' 
17.20 * 
3 
£_ 4 . 
7.20 * 
2B 
3 C 
+ 
4 d 
40 
1 1.20 
X5. 20 
19.20 ’ 
9 
1 
+ 
2B 
3 C 
• + 
4 d 
9.40 
&c. 
&c. 
&c. 
The arithmetical operation by the new series is undoubtedly 
easier than by the original series ; yet it is evident, by inspec- 
tion, that half the number of divisions by 20, (although easy 
operations), in the first and second series, may be exchanged 
for divisions by 10, which are no more than so many removals 
of the decimal point ; and that, in the third series, half the 
number of divisions by 40, (the first excepted) may be ex- 
changed for easier ones, one-fourth of them for divisions by 
20, and the other fourth for divisions by 10. The new series 
then, still converging somewhat quicker than by the powers 
of 80, may stand thus : 
and 
40 
5.20 
+ 
9. 10 
1 3 . 20 
+ 
17. 10 * 
3 
A 
+ 
B 
3 C 
1 
2D 
40 
7.20 
1 1 . 10 
15.20 
19.10 ’ 
9 
A 
+ • 
B 
3C 
+ 
D 
O TO > 
&c. 
&c. 
&c. 
And even yet one might still facilitate the computation of 
the value of some of the terms. Thus, — is 
1 + 
40 
is 
— 1 ~ t 
40 
IS 
= — : and 
15.20 
is = 
1 
100 : 
&c. 
By these expedients the sum of the three new series, which 
is equal to the hyperbolic logarithm of 10, may quickly be 
found. 
