Elementary arithmetic, with brief notices of its history... by Robert Potts.

14

Let inm, m, be the mantissse of u, u +1 respectively, and mn the
mantissa of iu + -o a being the sixth digit.
10
Now since tt, u + 1 and u + -,p are supposed to have the same
number of integral places, the characteristics of these logarithms will
be the same. Let this be c.
Then c + gl = log1o iu, c + n2 = log1o (zi + 1),
and o + i logilot + -
M - M=logio { U + - } ~-logi0luog { 1 - - }
Io~,10 1 q-l- '
=ilo CT0 loce +
log e10 ge{ 1 +
logetO { i07V - ( ~()) &c+. }
1 g l. nearly (I)
loglO 10 IO
- 1- loge 1
logel 1+0-
I 1 1 e.
= log1 0' 66 22 6 3 - - -
- 1o - 1 nearly (I
log10O 'i
vn - in,1 a
-12 - nl 10
From the logarithm tables of numbers consisting of five digits, which are
calculated to seven places of decimals, the logarithms of numbers consisting of six
digits may be found correctly as far as the seventh place of decimals.
In the investigation, the second and all succeeding terms of the series for ne -m
and n,2 — 21 have been omitted; and it must be shewn that these omissions do not
affect the seventh place of decimals in the logarithms.
21obr01 r 2zs ) of the series for rn- ac
First to shew that the second term L. l a 2 of the series for m-m
2 lo - 0 -l-' ~ 1J -
may be omitted.
1 1 1 1 2
Since <T- or -
Since 2loge10 2X2302... 4 '5 91
and the least value of au is 10000, and the greatest value of - is 1_
l; 10000'
also ca can never exceed 9.
Therefore 2 * 2 }is less than 2 X '0000000081 or '000000018.
lo21 10 is lessIU 9
1 1 2
And 2log.-10 is less than X -00000001 or 000000002.
21ogl0 'u2 9
Whence, it appears that the second terms of the two series do not affect the
seventh place of decimals of the logarithm, and therefore may be omitted, and ca
fortiori may the third and succeeding terms be omitted.