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

15

And ==m+(m2-t)-m - a of which (m2-mn).- is the increment
10 10
to be added to eni; and it is called "the proportional part," for
a sixth digit of the number, being a fourth proportional to 10,
a and n2h - mn.
Con. If the given number consisted of seven instead of six digits,
a being the sixth and b the seventh digit:
a b 6) a ~
then )n +(2 i) { + } = = + (n- MI). + - e) ~ - ),
-+0 1m 0 1 1
1 6
in which expression, 1 (m2i- ) b is the increment to be added for
the seventh digit, which is one-tenth of what should be added, if b
were the sixth digit.
18. PnoP. And conversely, to find the natural number correspond'ing to a
given logarithm, having given the mantissa of the logarithm of the number
composed of the first five digits, and the mantissa of the logarithm of the
next greater number than that which is composed of its first five digits.'
Employing the same notations, it has been proved that
nz -i nl a
qfn - en, 10
and since e, n, 2, are given, it follows that is known from
ad since,,, are ivn, it follo tha1,1 -,and the sixth digit a is known, and u + - is the number
n1 - 2 ~ 10
required.
To explain the method of constructiZng tables of proportional parts for a
table of logarithms calculated for all mntebers zwhich do not exceed five
digits; and their uses (1) in finding, by addition only, the logarithms
of numbers consisting of one or two digits more than the numbers for wzhich
the tables are calculated; and (2) in finding, by subtraction only, the
numbers corresponding to given logarithmns not found exactly in the tables.2
1 See note on p. 13.
2 Ex. 1. Find the logarithm of 43712'86.
Here loglo 43712=4-6406007 from the Tables
79 = prop. part for 8
5,9=,,, 6.'. logio 43712-86-=46406092
Ex. 2. Conversely. Find the natural number corresponding to the logarithm
4-6406092.
The given logarithm is = 4'6406092
logto 43712 = 4'6406007 from the Tables
85
79 = prop. part for 8
60
59,,, 6
1
Hence the numbecr corresponding to the logarithm. 4'6406092 is 43712'86.