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

16
If m, be the mantissa of the logarithm of a number of five digits
next greater than,, and mn, of that next less than m the mantissa of a
number whose sixth digit is a: it has been proved generally that the
nmantissa of the logarithm of the number of six digits is
is = 1, + ($, - mI) a, of which (2 - mn,) a is the increment or
10 10
the proportional part to be added to the mantissa of the logarithm
of the next less number of five digits.
The method will be best shewn by an example.
To construct a table of proportional parts for the logarithms of the
numbers between 43712 and 43713, consisting of six digits, having:>Lven logi 43712 = 4-6406007 and logot 43723 — 46406106.
Here,= -6406106, m, =-6406007,.'. z2-nl= '0000099.
'or a the sixth digit write 1, 2, 3, 4, 5, 6, 7, 8, 9, successively:a
a t (mt - mnll) '10
1 -00000099 = 0000010 nearly
2 ~00000198 = 0000020
3 -00000297 = 0000030
4 -00000396 = 0000040,,
5 -00000495 = 0000050,,
6 00000594 = 00000059
7 -00000693 = -0000069,,
8 -00000792 = 0000079
9 -00000891 = -0000089,,
In the printed tables of logarithms in general
use, the ciphers in the results above found are D 99
omitted, and only the significant figures of the
difference of the logarithms and the proper- 10
tional parts are registered on the right of each 2 20
figure which may occupy the place of the sixth 3 30
digit of the number. For the logarithms whose 4 40
difference is 99, the table of proportional parts 5 50
is given as here exhibited. 6 59
Since any digit in the seventh place of a number, is one-tenth of the same digit if it were in the
sixth place, it follows that tables of proportional 8 79
parts for digits in the seventh place will be one- 9 89
tenth of what each would be in the sixth place;
and consequently that separate tables for the proportioned parts for
the seventh digits will be unnecessary.