14,0 Mr. Hellins's Improvement in the Computation 
P. S. Containing an Improvement of Mr. Emerson's Computation 
of the Hyperbolic Logarithm of 10. 
7. Since the above paper was written, on looking into 
Emerson's Fluxions, I have found, at p. 137 of the first edi- 
tion,* another computation of the hyperbolic logarithm of 10,. 
which is preferable to Mr. Jones’s, on account of the swifter 
convergency of one of the series used in it, as will appear 
presently. These series also admit of a transformation to 
others, by which the constant divisors 81 and 64009, used 
by Mr. Emerson, are exchanged for 40 and 32000, while 
nearly the same rate of convergency is retained; which is 
another remarkable instance of the utility of transformations 
of this kind. 
Mr. Emerson, considering the number 10 as composed of 
4*° x k) 9 == 4^ x ^§1 ’ anc * lls * n g *h e same algebraic series as 
Mr. Jones used on this occasion, finds the hyperbolic loga- 
rithm of 10 to be = 10 L. of — + q L. of 
4 1 O IOOO 7 
+ 
l+£+ 
20 
J-9- 81 
i8-9 
3.253.64009 
+ 
+ 
20 
5-9- 81 * 

5. 25 3. 64009* 
+ 
+ 
7.9.81 3 
1 8 9 ^ 
7. 25 3 64009 
&C. 
r. &c. 
where, instead of a series converging by the powers of i, f 
as in Mr. Jones's calculation, we have that which converges by 
the powers of 
or above four times as swiftly. 
But what 
renders this very swiftly converging series still more useful is, 
that it admits of a transformation, by the theorem in article 4, 
* See also page 197 of 3d edition. 
f See article 3. 
