NEW SLIDE RULES—FIFTH ROOTS, ETC.—THE SOLUTION OF ALGEBRAIC
EQUATIONS—GAUGE POINTS AND SIGNS ON SLIDE RULES—TABLES AND DATA—SLIDE
RULE DATA SLIPS.
THE PICKWORTH SLIDE RULE.—In this rule, made by Mr. A. W. Faber, the
novel feature is the provision of a scale of cubes (F) in the stock or
body of the rule. From Fig. 33 it will be seen that the scale is fixed
on the bevelled side of a slotted recess in the back of the rule. The
slide carries an index mark, which is seen through the slot and can be
set to any graduation of the scale; in its normal position it agrees
with 1 on the scale. The C scale on the face of the rule is divided into
three equal parts by two special division lines, marked II. and III.,
which, together with the initial graduation 1 of the scale, serve for
setting or reading off values on the D scale. Similar division lines are
marked on the D scale.
[Illustration: FIG. 33.]
In using the rule for cubes or cube roots the slide is drawn to the
right, this movement never exceeding one-third of the length of the D
scale. With this limited movement, and with a single setting of the
slide, the values of ∛_̅a_, ∛(_a_ × 10), and ∛(_a_ × 100)) (_a_ being
less than 10 and not less than 1) are given simultaneously and without
any uncertainty as to the scales to use or the values to be read off.
_To Find the Cube of a Number._—The marks II. and III. on D divide that
scale into three equal sections. If the number to be cubed is in the
first section, I. on C is set to it; if in the second section, II. on C
is set to it; if in the third section, III. on C is set to it. Then,
under the index mark on the back of the slide will be found the
significant figures of the cube on the scale F. If I. on C was used for
the setting, the cube contains 1 digit; if II. was used, 2 digits; if
III. was used, 3 digits. If the first figure of the number to be cubed
is not in the units place, the decimal point is moved through _n_ places
so as to bring the first significant figure into the units place, the
cube found as above, and the decimal point moved in the _reverse
direction_ through 3_n_ places.
_To Find the Cube Root of a Number._—The index mark is set to the
significant figures of the number on scale F, and the cube root is read
on D under I., II. or III. on C, according as the number has 1, 2 or 3
digits preceding the decimal point. Numbers which have 1, 2 or 3 figures
preceding the decimal point are dealt with directly. Numbers of any
other form are brought to one of the above forms by moving the decimal
point 3 places (or such multiple of 3 places as may be required), the
root found and its decimal point moved 1 place for each 3-place
movement, but in the _reverse direction_.
THE “ELECTRO” SLIDE RULE.—In this special rule for electrical
calculations, made by Mr. A. Nestler, the upper scales run from 0·1 to
1000, and are marked “Amp.” and “sq. mm.” respectively. The lower scale
on the slide running from 1 to 10,000 is marked M (metres), while the
lower scale on the rule (0·1 to 100) is marked “Volt.” The latter scale
is so displaced that 10 on M agrees with 0·173 on the Volt scale. The
four factors involved are the current strength (in Amp.); the area of a
conductor (in sq. mm.); the length of the conductor (in metres); and the
permissible loss of potential (in volts). Having given any three of
these, the fourth can be found very readily. On the back of the slide
are a scale of squares, a scale of cubes and a single scale
corresponding to the D scale of an ordinary rule. Hence, by reversing
the slide, it is possible to obtain the 2nd, 3rd and 4th powers and
roots of numbers. In another form of the rule, the scale of metres is
replaced by one of yards, while instead of the area of the conductor in
sq. mm., the corresponding “gauge” sizes of wires are given.
THE “POLYPHASE” SLIDE RULE.—This instrument, made by the Keuffel & Esser
Company, New York, has, in addition to the usual scales, a scale of
cubes on the vertical edge of the stock of the rule, while in the centre
of the slide there is a reversed C scale; _i.e._, a scale exactly
similar to an ordinary C scale but with the graduations running from
right to left. The rule is specially useful for the solution of problems
containing combinations of three factors and problems involving squares,
square roots, cubes, cube roots and many of the higher powers and roots.
It is specially adapted for electrical and hydraulic work.
THE LOG-LOG DUPLEX SLIDE RULE.—The same makers have introduced a log-log
duplex slide rule, in which the log-log scale is in three sections,
placed one above the other, these occupying the position usually taken
up by the A scale. These scales are used in the manner already described
(page 86), but some advantage is obtained by the manner in which the
complete log-log scale is divided, the limits being _e_^{¹⁄₁₀₀} to _e_^⅒
(on Scale L.L. 1); _e_^⅒ to _e_ (on Scale L.L. 2); and _e_ to _e_^{10}
(on Scale L.L. 3), _e_ being the base of natural or hyperbolic
logarithms (2·71828). In this way a total log-log range of from 1·01 to
22,000 is provided, meeting all practical requirements. These log-log
scales are read in conjunction with a C scale placed at the upper edge
of the slide. A similar C scale, but reversed in direction, is placed at
the lower edge of the slide, this having red figures to distinguish it
readily. The adjacent scale on the body of the rule is an ordinary D
scale, and under this is an equally-divided scale giving the common
logarithms of values on D. In the centre of the slide is a scale of
tangents.
It will be understood that a “duplex” rule consists of two side strips
securely clamped together at the two ends, forming the body of the rule,
the slide moving between them; hence both front and back faces of the
rule and slide are available, graduations on the one side being referred
to those on the other by the cursor which extends around the whole. In
this instrument, the scales on the back face are the ordinary scales of
the standard rule with the addition of a scale of sines which is placed
in the centre of the slide. It will be evident that this instrument is
capable of dealing with a very wide range of problems involving
exponential and trigonometrical formulæ.
SMALL SLIDE RULES WITH MAGNIFYING CURSORS.—Several makers now supply 5
in. rules having the full graduations of a 10 in. rule, and fitted with
a magnifying cursor (Fig. 34). This forms a compact instrument for the
pocket, but owing to the closeness of the graduations it is not usually
possible to make a setting of the slide without using the cursor. This,
of course, involves more movements than with the ordinary instrument. It
is also very necessary to use the magnifying cursor in a _direct_ light,
if accurate readings are to be obtained. If these slight inconveniences
are to be tolerated, the principle could be extended, a 10 in. rule
being marked as fully as a 20 in., and fitted with a magnifying cursor.
The author has endeavoured, but without success, to induce makers to
introduce such a rule.
The magnifying cursor, supplied by Messrs. A. G. Thornton, Limited, has
a lens which fills the entire cursor. It has a powerful magnifying
effect, and the change from the natural to the magnified reading is less
abrupt than with the semicircular lens.
[Illustration: FIG. 34.]
THE CHEMIST’S SLIDE RULE.—A slide rule, specially adapted for chemical
calculations, has been introduced recently by Mr. A. Nestler. In this
instrument the C and D scales are as usually arranged; but, in place of
the A and B scales, there are a number of gauge points or marks denoting
the atomic and molecular weights of the most important elements and
combinations. The scales on the back of the slide are similarly
arranged, so that by reversing the slide the operations can be extended
very considerably. The rule finds its chief use in the calculation of
analyses. Thus, to find the percentage of chlorine if _s_ grammes of a
substance have been used and the precipitate of Ag.Cl. weighs _a_
grammes, we have the equation, _x_ = (Cl.)/(Ag.Cl.) × (_a_)/(_s_).
Hence, the mark Ag.Cl. on the upper scale of the slide is set to the
mark Cl. on the upper scale of the rule, when under _a_ on the C scale
is found the quantity of chlorine on D. By setting the cursor to this
value and bringing _s_ on C to the cursor, the percentage required can
be read on C over 10 on D.
The rule is also adapted to the solution of various other chemical and
electro-chemical calculations.
THE STELFOX SLIDE RULE.—This rule, shown in Fig. 35, has a stock 5 in.
long, fitted with a 10 in. slide jointed in the middle of its length by
means of long dowels. By separating the parts the compactness of a 5 in.
rule is obtained. The upper scales on the rule and slide resemble the
usual A and B scales. The D scale on the lower part of the stock is in
two sections, the second portion being placed below the first, as shown
in the illustration. The centre scale on the slide corresponds to the
usual C scale, while on the lower edge of the slide is a similar scale,
but with the index (1) in the middle of its length. The arrangement
avoids the necessity of resetting the slide, as is sometimes necessary
with the ordinary rule, and in general it combines the accuracy of a 10
in. rule with the compactness of a 5 in. rule; but a more frequent use
of the cursor is necessary. This rule is made by Messrs. John Davis &
Son, Limited, Derby.
[Illustration: FIG. 35.]
ELECTRICAL SLIDE RULE.—Another rule by the same makers, specially useful
for electrical engineers, has the usual scales on the working edges of
the rule and slide, while in the middle of the slide is placed a scale
of cubes. A log-log scale in two sections is provided; the power
portion, running from 1·07 to 2, is found on the lower part of the
stock, and the upper portion, running from 2 to 10^3, on the upper part
of the stock. The uppermost scale on the stock is in two parts, of which
that to the left, running from 20 to 100 and marked “Dynamo,” gives the
efficiencies of dynamos; that on the right, running from 20 to 100 and
marked “Motor,” gives the efficiencies of electric motors. The lowest
scale on the stock, marked “Volt,” gives the loss of potential in copper
conductors. The ordinary upper scale on the stock is marked L (length of
lead) at the left, and KW (kilowatts) at the right; the ordinary upper
scale on the slide is marked A (ampères) and mm^2 (sectional area) at
the left, and HP (horse-power) at the right. Additional lines on the
cursor enable the electrical calculations to be made either in British
or metric units.
THE PICOLET CIRCULAR SLIDE RULE.—A simple form of circular calculator,
made by Mr. L. E. Picolet of Philadelphia, is shown in Fig. 36. It
consists of a base disc of stout celluloid on which turns a smaller disc
of thin celluloid. A cursor formed of transparent celluloid is folded
over the discs, and is attached so that the friction between the cursor
and the inner disc enables the latter to be turned by moving the former.
By holding both discs the cursor can be adjusted as required. The
adjacent scales run in opposite directions, so that multiplication and
division are performed as with the inverted slide in an ordinary rule.
The outer scale, which is two-thirds the length of the main scale,
enables cube roots to be found. Square roots are readily determined and
continuous multiplication and division conveniently effected. Modified
forms of this neatly made little instrument are also available.
[Illustration: FIG. 36.]
OTHER RECENT SLIDE RULES.—Among other special types of slide rule,
mention should be made of the _Jakin_ 10 in. rule for surveyors, made by
Messrs. John Davis & Son, Limited, Derby. By the provision of a series
of short subsidiary scales, the multiplication of a sine or tangent of
an angle by a number can be obtained to an accuracy of 1 in 10,000. The
_Davis-Lee-Bottomley_ slide rule, by the same makers, has special scales
provided for circle spacing. The division of a circle into a number of
equal parts, often required in spacing rivets, bolts, etc., and in
setting out the teeth of gearwheels, is readily effected by the aid of
this instrument. The _Cuntz_ slide rule is a very comprehensive
instrument, having a stock about 2¼ in. wide, with the slide near the
lower edge. Above the slide are eleven scales, referable to the main
scales by the cursor. These scales enable squares and square roots,
cubes and cube roots, and areas and circumferences of circles to be
obtained by direct reading. A much more compact instrument could be
obtained by removing one-half the scales to the back of the rule and
using a double cursor.
[Illustration: FIG. 37.]
In one form of 10 in. rule, supplied by Mr. W. H. Harling, London, the
body of the rule is made of well-seasoned cane, with the usual celluloid
facings. The rule has a metal back, enabling the fit of the slide to be
regulated. This backing extends the full length of the rule, openings
about 1 in. long being provided at each end, enabling the scales on the
back of the slide to be set with greater facility than is possible with
the notched recesses usually adopted. The author has long endeavoured,
but without success, to induce makers to fit windows of glass or
celluloid in place of the notched recesses. This would allow the
graduation of the S and T scales to be set more accurately, and enable
both to be used at each end of the rule—an advantage in certain
trigonometrical calculations. It would have the further advantage of
permitting each alternate graduation of the evenly-divided or logarithm
scale to be placed at opposite sides of one central line, enabling the
reading to be made more accurately and conveniently.
Many special slide rules have lately been devised for determining the
time necessary to perform various machine-tool operations and for
analogous purposes, while attention has again been given to rules for
calculating the weights of iron and steel bars, plates, etc.
THE DAVIS-STOKES FIELD GUNNERY SLIDE RULE.—This rule, which is adapted
for calculations involved in “encounter” and “entrenched” field gunnery,
is designed for the 18 pr. quick-firing gun. The upper and lower
portions of the boxwood stock are united by a flexible centre of
celluloid, thus providing grooves front and rear to receive boxwood
slides. Each of the nineteen scales is marked with its name, and
corresponding scales are coloured red or black. The front edge is
bevelled and carries a scale of 1 in 20,000. The rule solves
displacement problems, map angles of sight, changes of corrector and
range corrections for changes in temperature, wind and barometer, etc. A
special feature for displacement calculations is the provision of a 50
yd. sub-base angle scale, by which the apex angle is read at one
setting.
THE DAVIS-MARTIN WIRELESS SLIDE RULE.—In wireless telegraphy it is
frequently necessary to determine wave-length, capacity or
self-induction when one or other of the factors of the equation, λ =
59·6√(LC) is unknown. The Davis-Martin wireless rule is designed to
simplify such calculations. The upper scale in the stock (inductance)
runs from 10,000 to 1,000,000; the adjacent scale on the slide
(capacity) runs from 0·0001 to 0·01 but in the reverse direction. The
lower scale on the stock (wave-length) runs from 100 to 1000, giving
square roots of the upper scale; while on the lower edge of the scale
are several arrows to suit the various denominations in which the
wave-length and capacity may be expressed.
IMPROVED CURSORS.—In some slide-rule operations, notably in those
involved in solving quadratic and cubic equations, it not infrequently
happens that readings are obscured by the frame of the cursor. Frameless
cursors have been introduced to obviate this defect. A piece of thick
transparent celluloid is sometimes employed, but this is liable to
become scratched in use. Fig. 37 shows a recent form of frameless glass
cursor made by the Keuffel & Esser Company, Hoboken, N.J., which is
satisfactory in every way.
Cursors having three hair lines are now fitted to some rules, the
distance apart of the lines being equal to the interval 0·7854–1 on the
A scale.
THE DAVIS-PLETTS SLIDE RULE.—In this rule a single log.-log. scale and
its reciprocal scale are arranged opposite the ordinary upper log.
scale. Thus, common logarithms can be read directly, while by taking
advantage of the properties of characteristics and mantissas of common
logarithms, the scale can be extended indefinitely. As 10 is the highest
number on the log.-log. scale, it is carried down to within 0·025 of
unity. The reading of log.-log. values above 10 is effected in a very
simple manner. There is also a scale in the centre of the slide which,
used in conjunction with the upper log. scale enables the natural
logarithm of any number between 0·0001 and 10,000 to be read direct,
while any number on the upper log. scale can be multiplied or divided by
_e^x_ if the latter is between these limits. On the back of the slide
are scales for all circular and hyperbolic functions, these being used
in conjunction with the upper log. scales.
THE CROMPTON-GALLAGHER BOILER EFFICIENCY CALCULATOR has a stock in the
thickness of which is a slot admitting a chart which can be moved at
right angles to the two separate slides. On the bevelled edge of one
slide, the graduations are continued so as to read against curves on the
chart, through an opening in the stock.
THE DAVIS-GRINSTED COMPLEX CALCULATOR.—This slide rule is of
considerable service in connection with calculations involving the
conversion of complex quantities from the form _a_ + _j_ _b_ to the form
R∠θ, and _vice versa_. The usual process of conversion necessitates
repeated reference to trigonometrical tables, and is both tedious and
time-taking. The Complex Calculator enables the conversions to be
effected without reference to tables and with the minimum expenditure of
time and labour.
The rule, which is about 16 in. long, has five scales. The upper one (A)
is an ordinary logarithmic scale thrice-repeated. The adjacent scales on
the slide comprise (1) a logarithmic scale of tangents (B) ranging from
0·1° to 45°, and (2) a logarithmic scale of secants (C) from 0° to 45°.
The lower scales D and E are identical with the A scale, and are
provided to enable multiplication, etc., to be performed without the
need for a separate slide rule. Readings can be transferred from A to
the lower scales by means of the cursor.
In using the rule to convert _a_ + _j_ _b_ to R∠θ, the index (45°) of
the B scale is set to the larger component and the cursor to the smaller
component, on scale A. Then θ (or its complement if _b_ is greater than
_a_) is read on B under the cursor. The cursor is then set to θ on the C
scale, and R is read on A under the cursor. The rule is made by Messrs.
John Davis & Son, Limited, Derby.