In addition, to the new forms of log.-log. slide rules previously
described, several other arrangements have been recently introduced,
notably a series by Mr. A. Nestler, of Lahr (London: A. Fastlinger, Snow
Hill). These comprise the “Rietz,” the “Precision,” the “Universal,” and
the “Fix” slide rules.
THE RIETZ RULE.—In this rule the usual scales A, B, C, and D, are
provided, while at the upper edge is a scale, which, being three times
the range of the D scale, enables cubes and cube roots to be directly
evaluated and also _n_^{³⁄₂} and _n_^⅔.
A scale at the lower edge of the rule gives the mantissa of the
logarithms of the numbers on D.
THE PRECISION SLIDE RULE.—In this rule the scales are so arranged that
the accuracy of a 20 in. rule is obtainable in a length of 10 in. This
is effected by dividing a 20 in. (50 cm.) scale length into two parts
and placing these on the working edges of the rule and slide. On the
upper and lower margins of the face of the rule are the two parts of
what corresponds to the A scale in the ordinary rule; while in the
centre of the slide is the scale of logarithms which, used in
conjunction with the 50 cm. scales on the slide, is virtually twice the
length of that ordinarily obtainable in a 10 in. rule. The same remark
applies to the trigonometrical scales on the under face of the slide.
Both the sine and tangent scales are in two adjacent lengths, while on
the edge of the stock of the rule, below the cursor groove, is a scale
of sines of small angles from 1° 49′ to 5° 44′. This is referred to the
50 cm. scales by an index projection on the cursor.
If C and C′ are the two parts of the scale on the slide and D and D′ the
corresponding scales on the rule, it is clear that in multiplying two
factors 1 on C can only be set directly to the upper scale D; while 10
on C′ can only be set directly to the lower scale D′. Hence if the first
factor is greater than about 3·2, the cursor must be used to bring 1 on
C to the first factor on D′. Similarly, in division, numerators and
denominators which occur on C and D′ or on C′ and D cannot be placed in
direct coincidence but must be set by the aid of the cursor.
Any uncertainty in reading the result can be avoided by observing the
following rule: _If in setting the index_ (1 _or_ 10) _in
multiplication, or in setting the numerator to the denominator in
division, it is necessary to cross the slide, then it will also be
necessary to cross the slide to read the product or quotient._
THE UNIVERSAL SLIDE RULE.—In this instrument the stock carries two
similar scales running from 1 to 10, to which the slide can be set.
Above the upper one is the logarithm scale and under the lower one the
scale of squares 1 to 100. On the edge of the stock of the rule, under
the cursor groove, is a scale running from 1 to 1000. An index
projecting from the cursor enables this scale to be used with the scales
on the face of the rule, giving cubes, cube roots, etc.
On the slide, the lower scale is an ordinary scale, 1 to 10. The centre
scale is the first part of a scale giving the values of sin _n_ cos _n_,
this scale being continued along the upper edge of the slide (marked
“sin-cos”) up to the graduation 50. On the remainder of this line is a
scale running from right to left (0 to 50) and giving the value of
cos^2_n_. In surveying, these scales greatly facilitate the calculations
for the horizontal distance between the observer’s station and any
point, and the difference in height of these two points.
On the back of the slide are scales for the sines and tangents of
angles. The values of the sines and tangents of angles from 34′ to 5°
44′ differ little from one another, and the one centre scale suffices
for both functions of these small angles.
THE FIX SLIDE RULE.—This is a standard rule in all respects, except that
the A scale is displaced by a distance (π)/(4) so that over 1 on D is
found 0·7854 on A. This enables calculations relating to the area and
cubic contents of cylinders to be determined very readily.
THE BEGHIN SLIDE RULE.—We have seen that a disadvantage attending the
use of the ordinary C and D scales, is that it is occasionally necessary
to traverse the slide through its own length in order to change the
indices or to bring other parts of the slide into a readable position
with regard to the stock. To obviate this disadvantage, Tserepachinsky
devised an ingenious arrangement which has since been used in various
rules, notably in the Beghin slide rule made by Messrs. Tavernier-Gravêt
of Paris. In this rule the C and D scales are used as in the standard
rule, but in place of the A and B scales, we have another pair of C and
D scales, displaced by one-half the length of the rule. The lower pair
of scales may therefore be regarded as running from 10^{_n_} to 10^{_n_
+ 1}, and the upper pair as running from √(10) × 10^{_n_} to √(10) ×
10^{_n_ + 1}. With this arrangement, _without moving the slide more than
half its length_, to the left or right, it is always possible to compare
_all values between_ 1 _and_ 10 _on the two scales_. This is a great
advantage especially in continuous working.
Another commendable feature of the Beghin rule is the presence of a
reversed C scale in the centre of the slide, thus enabling such
calculations as _a_ × _b_ × _c_ to be made with one setting of the
slide. On the back of the slide are three scales, the lowest of which,
used with the D scale, is a scale of squares (corresponding to the
ordinary B scale), while on the upper edge is a scale of sines from 5°
44′ to 90°, and in the centre, a scale of tangents from 5° 43′ to 45°.
On the square edge of the stock, under the cursor groove, is the
logarithm scale, while on the same edge, above the cursor groove, are a
series of gauge points. All these values are referred to the face scales
by index marks on the cursor.
THE ANDERSON SLIDE RULE.—The principle of dividing a long scale into
sections as in the Precision rule, has been extended in the Anderson
slide rule made by Messrs. Casella & Co., London, and shown in Fig. 17.
In this the slide carries a scale in four sections, used in conjunction
with an exactly similar set of scale-lines in the upper part of the
stock. On the lower part of the stock is a scale in eight sections
giving the square roots of the upper values. In order to set the index
of the slide to values in the stock, two indices of transparent
celluloid are fixed to the slide extending over the face of the rule as
shown in the illustration. As each scale section is 30 cm. in length,
the upper lines correspond to a single scale of nearly 4 ft., and the
lower set to one of nearly 8 ft. in length, giving a correspondingly
large increase in the number of subdivisions of these scales, and
consequently much greater accuracy.
In order to decide upon which line a result is to be found, sets of
“line numbers” are marked at each end of the rule and slide and also on
the metal frame of the cursor. In multiplication, the line number of the
product is the sum of the line numbers of the factors if the left index
is used, or 1 more than this sum if the right index is used. The
illustration shows the multiplication of 2 by 4. The left index is set
to 2 (line number, 1), and the cursor set to 4 on the slide (line
number, 2); hence, as the left index is used, the result is found on
line No. 3. Similar rules are readily established for division. The
column of line numbers headed 0 is used for units, that headed 4 for
tens, and so on; one column is given for tenths, headed −4. The square
root scale bears similar line numbers, so that the square root of any
value on the upper scales is found on the correspondingly figured line
below.
[Illustration: FIG. 17.]
THE MULTIPLEX SLIDE RULE differs from the ordinary form of rule in the
arrangement of the B scale. The right-hand section of this scale runs
from left to right as ordinarily arranged, but the left-hand section
runs in the reverse direction, and so furnishes a reciprocal scale. At
the bottom of the groove, under the slide, there is a scale running from
1 to 1000, which is used in conjunction with the D scale, readings being
referred thereto by a metal index on the end of the slide. By this means
cubes, cube roots, etc., can be read off directly. Messrs. Eugene
Dietzgen & Co., New York, are the makers.
THE “LONG” SLIDE RULE has one scale in two sections along the upper and
lower parts of the stock, as in the “Precision” rule. The scale on the
slide is similarly divided, but the graduations run in the reverse
direction, corresponding to an inverted slide. Hence the rules for
multiplication and division are the reverse of those usually followed
(page 30). On the back of the slide is a single scale 1–10, and a scale
1–1000, giving cubes of this single scale. By using the first in
conjunction with the scales on the stock, squares may be read, while in
conjunction with the cube scale, various expressions involving squares,
cubes and their roots may be evaluated.
HALL’S NAUTICAL SLIDE RULE consists of two slides fitting in grooves in
the stock, and provided with eight scales, two on each slide, and one on
each edge of each groove. While fulfilling the purposes of an ordinary
slide rule, it is of especial service to the practical navigator in
connection with such problems as the “reduction of an ex-meridian sight”
and the “correction of chronometer sights for error in latitude.” The
rule, which has many other applications of a similar character, is made
by Mr. J. H. Steward, Strand, London.