Although the 10 in. slide rule is probably the most serviceable form of
calculating instrument for general purposes, many prefer the more
portable circular calculator, of which many varieties have been
introduced during recent years. The advantages of this type are: It is
more compact and conveniently carried in the waistcoat pocket. The
scales are continuous, so that no traversing of the slide from 1 to 10
is required. The dial can be set quickly to any value; there is no
trouble with tight or ill-fitting slides. The disadvantages of most
forms are: Many problems involve more operations than a straight rule.
The results being read under fingers or pointers, an error due to
parallax is introduced, so that the results generally are not so
accurate as with a straight rule. The inner scales are short, and
therefore are read with less accuracy. Special scale circles are needed
for cubes and cube roots. The slide cannot be reversed or inverted.
[Illustration: FIG. 22.]
[Illustration: FIG. 23.]
THE BOUCHER CALCULATOR.—This circular calculator resembles a
stem-winding watch, being about 2 in. in diameter and ⁹⁄₁₆in. in
thickness. The instrument has two dials, the back one being fixed, while
the front one, Fig. 22 (showing the form made by Messrs. W. F. Stanley,
London), turns upon the large centre arbor shown. This movement is
effected by turning the milled head of the stem-winder. The small centre
axis, which is turned by rotating the milled head at the side of the
case, carries two fine needle pointers, one moving over each dial, and
so fixed on the axis that one pointer always lies evenly over the other.
A fine index or pointer fixed to the case in line with the axis of the
winding stem, extends over the four scales of the movable dial as shown.
Of these scales, the second from the outer is the ordinary logarithmic
scale, which in this instrument corresponds to a straight scale of about
4¾in. in length. The two inner circles give the square roots of the
numbers on the primary logarithmic scale, the smaller circle containing
the square roots of values between 1 and 3·162 (= √(10)), while the
other section corresponds to values between 3·162 and 10. The outer
circle is a scale of logarithms of sines of angles, the corresponding
sines of which can be read off on the ordinary scale.
On the fixed or back dial there are also four scales, these being
arranged as in Fig. 23. The outer of these is a scale of equal parts,
while the three inner scales are separate sections of a scale giving the
cube roots of the numbers taken on the ordinary logarithmic scale and
referred thereto by means of the pointers. In dividing this cube-root
scale into sections, the same method is adopted as in the case of the
square-root scale. Thus, the smallest circle contains the cube roots of
numbers between 1 and 10, and is therefore graduated from 1 to 2·154;
the second circle contains the cube roots of numbers between 10 and 100,
being graduated from 2·154 to 4·657; while the third section, in which
are found the cube roots of numbers between 100 and 1000, carries the
graduations from 4·657 to 10.
What has been said in an earlier section regarding the notation of the
slide rule may in general be taken to apply to the scales of the Boucher
calculator. The manner of using the instrument is, however, not quite so
evident, although from what follows it will be seen that the operative
principle—that of variously combining lengths of a logarithmic scale—is
essentially similar. In this case, however, it is seen that in place of
the straight scale-lengths shown in Fig. 4, we require to add or
subtract arc-lengths of the circular scales, while, further, it is
evident that in the absence of a fixed scale (corresponding to the stock
of the slide rule) these operations cannot be directly performed as in
the ordinary form of instrument. However, by the aid of the fixed index
and the movable pointer, we can effect the desired combination of the
scale-lengths in the following manner. Assuming it is desired to
multiply 2 by 3, the dial is turned in a backward direction until 2 on
the ordinary scale lies under the fixed index, after which the movable
pointer is set to 1 on the scale. As now set, it is clear that the
arc-length 1–2 is spaced off between the fixed index and the movable
pointer, and it now only remains to add to this definite arc-length a
further length of 1–3. To do this we turn the dial still further
backward until the arc 1–3 has passed under the movable pointer, when
the result, 6, is read under the fixed index. A little consideration
will show that any other scale length may be added to that included
between the fixed and movable pointers, or, in other words, any number
on the scale may be multiplied by 2 by bringing the number to the
movable pointer and reading the result under the fixed index. The rule
for multiplication is now evident.
_Rule for Multiplication._—_Set one factor to the fixed index and bring
the pointer to 1 on the scale; set the other factor to the pointer and
read the result under the fixed index._
With the explanation just given, the process of division needs little
explanation. It is clear that to divide 6 by 3, an arc-length 1–3 is to
be taken from a length 1–6. To this end we set 6 to the index
(corresponding in effect to passing a length 1–6 to the left of that
reference point) and set the pointer to the divisor 3. As now set, the
arc 1–6 is included between 1 on the scale and the index, while the arc
1–3 is included between 1 on the scale and the pointer. Obviously if the
dial is now turned forward until 1 on the scale agrees with the pointer,
an arc 1–3 will have been deducted from the larger arc 1–6, and the
remainder, representing the result of this operation, will be read under
the index as 2.
_Rule for Division._—_Set the dividend to the fixed index, and the
pointer to the divisor; turn the dial until 1 on the scale agrees with
the pointer, and read the result under the fixed index._
The foregoing method being an inversion of the rule for multiplication,
is easily remembered and is generally advised. Another plan is, however,
preferable when a series of divisions are to be effected with a constant
divisor—_i.e._, when _b_ in (_a_)/(_b_) = _x_ is constant. In this case
1 on the scale is set to the index and the pointer set to _b_; then if
any value of a is brought to the pointer, the quotient _x_ will be found
under the index.
_Combined Multiplication and Division_, as (_a_ × _b_ × _c_)/(_m_ × _n_)
= _x_, can be readily performed, while cases of continued multiplication
evidently come under the same category, since _a_ × _b_ × _c_ = (_a_ ×
_b_ × _c_)/(1 × 1) = _x_. Such cases as _a_/(_m_ × _n_ × _r_) = _x_ are
regarded as (_a_ × 1 × 1 × 1)/(_m_ × _n_ × _r_) = _x_; while (_a_ × _b_
× _c_)/(_m_) = _x_ is similarly modified, taking the form (_a_ × _b_ ×
_c_)/(_m_ × 1) = _x_. In all cases the expression must be arranged so
that there is _one more factor in the numerator_ than _in the
denominator_, _1’s being introduced as often as required_. The simple
operations of multiplication and division involve a similar disposition
of factors, since from the rules given it is evident that _m_ × _n_ is
actually regarded as (_m_ × _n_)/(1), while (_m_)/(_n_) becomes in
effect (_m_ × 1)/(_n_). It is important to note the general
applicability of this arrangement-rule, as it will be found of great
assistance in solving more complicated expressions.
As with the ordinary form of slide rule, the factors in such an
expression as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_ are taken in the
order:—1st factor of numerator; 1st factor of denominator; 2nd factor of
numerator; 2nd factor of denominator, and so on; the 1st factor as _a_
being set to the index, and the result _x_ being finally read at the
same point of reference.