Plane trigonometry with practical applications, by Leonard E. Dickson.

44

TRIGONOMETRY

[Art. 31

Hence multiplication and division can be performed geometrically
by means of a logarithmic scale (Fig. 23).
31. Slide rules. It was noticed by W. Oughtred in 1630 that
we may dispense with the use of the dividers in Art. 30 by using
two like logarithmic scales E and F (Fig. 24), which slide along
1E 3 4 5 6 7 8 9 10
F _ 3 4 5 6 7 8 910
FIG. 24
each other. The process to multiply 2 by 3 now consists in placing
scale F so that its point marked 1 is underneath the point marked
2 on scale E; then above the point marked 3 on scale F we find
on scale E the required product 6.
To divide 6 by 3, place scale F so that its point 3 falls under
point 6 on scale E and read on E the point above 1 on F.
The above pair of logarithmic scales enables us to find the
products, when not exceeding 10, of numbers having only one
digit, and hence would be of no practical use. But we readily
extend its usefulness as follows.
First, by using also log 1.1, log 1.2,..., log 9.9, we subdivide
each main interval in Fig. 23 into ten (unequal) divisions, which
are marked by short bars (and every fifth one by a longer bar)
and not by numbers (1.1, 1.2,..., 1.9 for the first main division).
Such a modified logarithmic scale is shown in the left half of
scale A in Fig. 25, where are omitted the subdivisions of the
shorter main divisions.
By using two such modified scales sliding along each other, as
the left halves of scales A and B in Fig. 25, we can find the products of numbers each having a units' digit and one decimal place.
Second, we can now readily extend the scale to include all
whole numbers not exceeding 100. For example, we obtain a
segment representing log 27 = 1 + log 2.7 by adding the length