It has been shown that the degree of accuracy attainable in slide-rule
calculations depends upon the length of scale employed. Considerations
of general convenience, however, render simple straight-scale rules of
more than 20 in. in length inadmissible, so that inventors of long-scale
slide rules, in order to obtain a high degree of precision, combined
with convenience in operation, have been compelled to modify the
arrangement of scales usually employed. The principal methods adopted
may be classed under three varieties: (1) The use of a long scale in
sectional lengths, as in Hannyngton’s Extended Slide Rule and Thacher’s
Calculating Instrument; (2) the employment of a long scale laid in
spiral form upon a disc, as in Fearnley’s Universal Calculator and
Schuerman’s Calculating Instrument; and (3) the adoption of a long scale
wound helically upon a cylinder, of which Fuller’s and the “R.H.S.”
Calculating Rules are examples.
FULLER’S CALCULATING RULE.—This instrument, which is shown in Fig. 18,
consists of a cylinder _d_ capable of being moved up and down and around
the cylindrical stock _f_, which is held by the handle. The logarithmic
scale-line is arranged in the form of a helix upon the surface of the
cylinder _d_, and as it is equivalent to a straight scale of 500 inches,
or 41 ft. 8 in., it is possible to obtain four, and frequently five,
figures in a result.
Upon reference to the figure it will be seen that three indices are
employed. Of these, that lettered _b_ is fixed to the handle; while two
others, _c_ and _a_ (whose distance apart is equal to the axial length
of the complete helix), are fixed to the innermost cylinder _g_. This
latter cylinder slides telescopically in the stock _f_, enabling the
indices to be placed in any required position relatively to _d_. Two
other scales are provided, one (_m_) at the upper end of the cylinder
_d_, and the other (_n_) on the movable index.
[Illustration: FIG. 18.]
In using the instrument a given number on _d_ is set to the fixed index
_b_, and either _a_ or _c_ is brought to another number on the scale.
This establishes a ratio, and if the cylinder is now moved so as to
bring any number to _b_, the fourth term of the proportion will be found
under _a_ or _c_. Of course, in multiplication, one factor is brought to
_b_, and _a_ or _c_ brought to 100. The other factor is then brought to
_a_ or _c_, and the result read off under _b_. Problems involving
continuous multiplication, or combined multiplication and division, are
very readily dealt with. Thus, calling the fixed index F, the upper
movable index A, and the lower movable index B, we have for _a_ × _b_ ×
_c_:—Bring _a_ to F; A to 100; _b_ to A or B; A to 100; _c_ to A or B
and read the product at F.
The maximum number of figures in a product is the sum of the number of
figures in the factors and this results when all the factors except the
first have to be brought to B. Each time a factor is brought to A, 1 is
to be deducted from that sum.
For division, as _a_/(_m_ × _n_), bring _a_ to F; A or B to _m_; 100 to
A; A or B to _a_; 100 to A and read the quotient at F.
[Illustration: FIG. 19.]
The maximum number of figures in the quotient is the difference between
the sum of the number of figures in the numerator factors and those of
the denominator factors, _plus_ 1 for each factor of the denominator and
this results when A has to be set to all the factors of the denominator
and all the factors of the numerator except the first brought to B. Each
time B is set to a denominator factor or a numerator factor is brought
to A, 1 is to be deducted.
Logarithms of numbers are obtained by using the scales _m_ and _n_ and
hence powers and roots of any magnitude may be obtained by the procedure
already fully explained. The instrument illustrated is made by Messrs.
W. F. Stanley & Co., Limited, London.
THE “R.H.S.” CALCULATOR.—In this calculator, designed by Prof. R. H.
Smith, the scale-line, which is 50 in. long, is also arranged in a
spiral form (Fig. 19), but in this case it is wrapped around the central
portion of a tube which is about ¾in. in diameter and 9½in. long. A
slotted holder, capable of sliding upon the plain portions of this tube,
is provided with four horns, these being formed at the ends of the two
wide openings through which the scale is read. An outer ring carrying
two horns completes the arrangement.
One of the horns of the holder being placed in agreement with the first
factor, and one of the horns of the ring with the second factor, the
holder is moved until the third factor falls under the same horn of the
ring, when the resulting fourth term will be found under the same (right
or left) horn of the holder, at either end of the slot. In
multiplication, 100 or 1000 is taken for the second factor in the above
proportion, as already explained in connection with Fuller’s rule;
indeed, generally, the mode of operation is essentially similar to that
followed with the former instrument.
The scale shown on one edge of the opening in the holder, together with
the circular scale at the top of the spiral, enables the mantissæ of
logarithms of numbers to be obtained, and thus problems involving powers
and roots may be dealt with quite readily. This instrument is supplied
by Mr. J. H. Steward, London.
THACHER’S CALCULATING INSTRUMENT, shown in Fig. 20, consists of a
cylinder 4 in. in diameter and 18 in. long, which can be given both a
rotary and a longitudinal movement within an open framework composed of
twenty triangular bars. These bars are connected to rings at their ends,
which can be rotated in standards fixed to the baseboard. The scale on
the cylinder consists of forty sectional lengths, but of each scale-line
that part which appears on the right-hand half of the cylinder is
repeated on the left-hand half, one line in advance. Hence each half of
the cylinder virtually contains two complete scales following round in
regular order. On the lower lines of the triangular bars are scales
exactly corresponding to those on the cylinder, while upon the upper
lines of the bars and not in contact with the slide is a scale of square
roots.
[Illustration: FIG. 20.]
By rotating the slide any line on it may be brought opposite any line in
frame and by a longitudinal movement any graduation on these lines may
be brought into agreement. The whole can be rotated in the supporting
standards in order to bring any reading into view. As shown in the
illustration, a magnifier is provided, this being conveniently mounted
on a bar, along which it can be moved as required.
SECTIONAL LENGTH OR GRIDIRON SLIDE RULES.—The idea of breaking up a long
scale into sectional lengths is due to Dr. J. D. Everett, who described
such a gridiron type of slide rule in 1866. Hannyngton’s Extended Slide
Rule is on the same principle. Both instruments have the lower scale
repeated. H. Cherry (1880) appears to have been the first to show that
such duplication could be avoided by providing two fixed index points in
addition to the natural indices of the scale. These additional indices
are shown at 10′ and 100′ in Fig. 21, which represents the lower sheet
of Cherry’s Calculator on a reduced scale. The upper member of the
calculator consists of a transparent sheet ruled with parallel lines,
which coincide with the lines of the lower scale when the indices of
both are placed in agreement. To multiply one number by another, one of
the indices on the upper sheet is placed to one of the factors, and the
position of whichever index falls under the transparent sheet is noted
on the latter. Bringing the latter point to the other factor, the result
is found under whichever index lies on the card. In other arrangements
the inventor used transparent scales, the graduations running in a
reverse direction to those of the lower scale. In this case, a factor on
the upper scale is set to the other factor on the lower, and the result
read at the available index.
[Illustration: FIG. 21.]
PROELL’S POCKET CALCULATOR is an application of the last-named
principle. It comprises a lower card arranged as Fig. 21, with an upper
sheet of transparent celluloid on which is a similar scale running in
the reverse direction. For continued multiplication and division, a
needle (supplied with the instrument) is used as a substitute for a
cursor, to fix the position of the intermediate results. A series of
index points on the lower card enable square and cube roots to be
extracted very readily. This calculator is supplied by Messrs. John J.
Griffin & Sons, Ltd., London.