Plane trigonometry, by S.L. Loney.

CHAPTER XI.

TABLES OF LOGARITHMS AND TRIGONOMETRICAL RATIOS.
PRINCIPLE OF PROPORTIONAL PARTS.
148. WE have pointed out that the logarithms of all
numbers from 1 to 108000 may be found in Chambers'
Mathematical Tables, so that, for example, the logarithms
of 74583 and 74584 may be obtained directly therefrom.
Suppose however we wanted the logarithm of a
number lying between these two, e.g. the number 74583'3.
To obtain the logarithm of this number we use the
Principle of Proportional Parts which states that the
increase in the logarithm of a number is proportional to
the increase in the number itself.
Thus from the tables we find
log 74583 = 4-8726398...............(1),
and log 74584 = 4-8726457...............(2).
The quantity log 74583'3 will clearly lie between
log 74583 and log 74584.
Let then log 74583'3 = log 74583 + x
= 4-8726398 + x............(3).