A simple method of obtaining powers and roots, which may serve on
occasion, is by scaling off proportional lengths on the D scale (or the
A scale) of the ordinary rule. Thus, to determine the value of
1·25^{1·67} we take the actual length 1–1·25 on D scale, and increase it
by any convenient means in the proportion of 1 ∶ 1·67. Then with a pair
of dividers we set off this new length from 1, and obtain 1·44 as the
result. One convenient method of obtaining the desired ratio is by a
pair of proportional compasses. Thus to obtain 1·52^{¹⁷⁄₁₆}, the
compasses would be set in the ratio of 16 to 17, and the smaller end
opened out to include 1–1·52 on the D scale; the opening in the large
end of the compasses will then be such that setting it off from 1 we
obtain 1·56 on D as the result sought.
[Illustration: FIG. 11.]
The converse procedure for obtaining the _n_th root of a number N will
obviously resolve itself into obtaining (1)/(_n_)th of the scale length
1-N, and need not be further considered.
Simple geometrical constructions are also used for obtaining scale
lengths in the required ratio. A series of parallel lines ruled on
transparent celluloid or stout tracing paper may be placed in an
inclined position on the face of the rule and adjusted so as to divide
the scale as desired. When much work is to be done which requires values
to be raised to some constant but comparatively low power, _n_, the
author has found the following device of assistance:—On a piece of thin
transparent celluloid a line OC is drawn (Fig. 11) and in this a point B
is taken such that (OC)/(OB) is the desired ratio. It is convenient to
make OB = 1–10 on the A scale, so that assuming we require a series of
values of _v_^{1·35}, OB would be 12·5 cm. and OC, 16·875 cm. On these
lines semi-circles are drawn as shown, both passing through the point O.
Applying this cursor to the upper scales so that the point O is on 1 and
the semi-circle O M B passes through _v_ on A, the larger semi-circle
will give on A the value of _v^n_. Thus for _p_ _v^n_ = 39·5 ×
4·9^{1·35}, set 1 on B to 39·5 on A (Fig. 12) and apply the cursor to
the working edge of B, so that O agrees with 1 and O M B passes through
4·9 on B. The larger semi-circle then cuts the edge of the slide on a
point, giving 337 on A as the result required.
Of course any number of semi-circles may be drawn, giving different
ratios. If a number of evenly-spaced divisions are used as bases, the
device affords a simple means of obtaining a succession of small powers
or roots, while it also finds a use in determining a number of geometric
means between two values as is required in arranging the speed gears of
machine tools, etc. The converse operation of finding roots will be
evident as will also many other uses for which the device is of service.
[Illustration: FIG. 12.]
The lines should be drawn in Indian ink with a very sharp pen and on the
_under_ side of the celluloid so that the lines lie in close contact
with the face of the rule.
_The Radial Cursor_, another device for the same purpose, is always used
in conjunction with the upper scales. As will be seen from Fig. 13, the
body of the cursor P carries a graduated bar S which can be removed in a
direction transverse to the rule, and adjusted to any desired position.
Pivoted to the lower end of S is a radial arm R of transparent celluloid
on which a centre line is engraved.
A reference to the illustration will show that the principle involved is
that of similar triangles, the width of the slide being used as one of
the elements. Thus, to take a simple case, if 2 on S is set to the index
on P, and 1 on B is brought to N on A, then by swinging the radial arm
until its centre line agrees with 1 on C, we can read N^2 on A.
Evidently, since in the two similar triangles A O N^2 and N _t_ N^2 the
length of A O is twice that of N _t_, it results that A N^2 = 2 A N. In
general, then, to find the _n_th power of a number, we set the cursor to
1 or 10 on A, bring _n_ on the cross bar S to the index on the cursor,
and 1 on B to N on A. Then to 1 on C we set the line on the radial arm,
and under the latter read N^{_n_} on A. The inverse proceeding for
finding the _n_th root will be obvious.
[Illustration: FIG. 13.]
An advantage offered by this and analogous methods of obtaining powers
and roots is that the result is obtained on the ordinary scale of the
rule, and hence it can be taken directly into any further calculation
which may be necessary.