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THE SLIDE RULE:
A PRACTICAL MANUAL
': -'•";'' • ' . ■ .'•■,- ' '■ ' ■' ■ '■' ' ' ■\'' /: r WS
C. N. PICKWORTH |
ADVERTISEMENTS
©
©
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© ®
$
NESTLER'S SLIDE RULES
OBTAINABLE FROM ALL DRAWINC OFFICE
MATERIAL DEALERS
STANDARD TYPE, in well-seasoned wood, with
celluloid facings screwed on. 5in., 8in., 10in.,
loin., 20in. and 24in.
THE RIETZ SLIDE RULE enables cubes and
cube roots to be read off directly and logarithms
to be found quickly and accurately without using
the slide. 10in., 15in. and 20in.
THE PRECISION SLIDE RULE— a 20in.
rule in a length of lOin. Gives results to within
0*03 per cent, and a corresponding accuracy with
the other scales.
THE UNIVERSAL SLIDE RULE, lOin.,
allows all the usual operations to be performed ;
gives cubes and cube roots directly, and enables
various tacheometrical calculations to be readily
effected.
THE FIX SLIDE RULE, lOin. All the usual
operations can be performed and the area and
cubic contents of round bodies determined in a
very simple manner.
CLOSELY DIVIDED Bin. POCKET RULE,
with magnifying cursor (full-size illustration on
side).
CONSTRUCTIONAL IMPROVEMENTS
ELASTIC STRIPS which are let into the sides of the
stock ensure smooth and even movement independent of
atmospherical Influences.
The stock, being covered with CELLULOID on both TOP
and HOTTOM, cannot warp, as wood and celluloid ten
differently affected by climatic changes.
ALL CELLULOID SCALES are screwed on to the
wood.
WHOLESALE DEPOT I
A. FASTLINGER
30 Snow Hill, LONDON, E.C.
ADVKKTISKMENTS
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. Manufacturers of . .^___ LdONTDON^
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. of All Descriptions .
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Curve Paper Log. Paper Semi = Log. Paper
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The Brunsviga CALCULATOR
Is not an adding machine, but it will do any calculations
such as are usual —for instance, in Cotton Mills,
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etc. etc. It will make up Length-notes, extend
by pence and pence fractions, in vulgar or deci-
mal fractions, calculate averages, percentages,
costs, wages, dividends, convert foreign
measures and currencies into English and
7>iceversa much quicker than any expert; in- >
fallibly accurate and without any brain fag.
Apply for free specimens of textile work that can be done by the BRUNSVIGA
A Few Examples :
12,435 yards @ 3^d. = ^17016s. 5d in 5 Seconds.
11,778 yards costal, 125— Cost per yard = 24"96d 10 ,,
1,010 yards («)3-663d. per 20 yards = ^0 15s. 5d 5
1,235 yards @ 1/5 per 40 yards = £'2 3h. 9d 3 ,,
Difference between ^465 and .£.595 = 27 -90% 10
5 /36* 3 /sn 9 /»» u /wh ls /m 8 /m 19 /m = 3 ' 250 yards - 1 Wh . ole .
3,250 yards @2*' = £88 0s. 2d. hSwnnT
less 2 \ = £32 3s. 8d. J 18 Seconds -
@ 25-15 into Francs 809*37
Apply for Booklet 32, showing specimen Calculations and Testimonials. Special Booklet for
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DYKES BROS., 4 Albert Sq., MANCHESTER
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and 75B Queen Victoria Street, "Hornsby House," LONDON, E.C.
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DRAWING
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JOHN DAVIS & SON f DERBY ), LD.
ALL SAINTS WORKS
. . . DERBY
BY THE SAME AUTHOR.
LOGARITHMS FOR BEGINNERS.
"An extremely useful and much-needed little work, giving
a complete explanation of the theory and use of logarithms,
by a teacher of great clearness and good style." — The Mining
Journal.
THE INDICATOR : ITS CONSTRUCTION
AND APPLICATION.
["Thk Indicator Handbook" : Part I.].
"A valuable little volume. . . . Mr Pickwortb's judg-
ment is always sound, and is evidently derived from a
personal acquaintance with indicator work." — The Engineer \
THE INDICATOR DIAGRAM: ITS
ANALYSIS AND CALCULATION.
["The Indicator Handbook": Part II.].
"An excellent guide to the intelligent interpretation and
use of the diagram. Of high value to the practical
engineer. " — Marine Engineering.
POWER COMPUTER
FOR STEAM, GAS, AND OIL ENGINES.
"Accurate, expeditious, and thoroughly practical. . . .
We can confidently recommend it, and engineers will find it
a great boon in undertaking tests, etc." — The Electrician.
0. N. PIOKWOEiTH, FALLOWFIELD, MANCHESTER.
THE
SLIDE EULE:
A PRACTICAL MANUAL.
CHARLES N. PICKWORTH
n
whitworth scholar; editor of " the mechanical world " ; author of
"logarithms for beginners"; "the indicator: its construction
and application"; "the indicator diagram: its analysis and
calculation," etc.
OF THE
UNIVERSITY
OF
*UFOBNli
ELEVENTH EDITION
Manchester :
Emmott and Co., Limited,
65 King Street;
and
Charles N. Pick worth,
Fallowfield.
London :
WHITTAKER AND Co.,
2 White Hart Street, E.C.
New York :
D. Van Nostrand Co.
23 Murray Street.
[all rights reserved]
4 73
etwtKAL
EDINBURGH
COLSTON AND COY. LIMITED
PRINTERS
PREFACE TO THE ELEVENTH EDITION.
HPHE Eleventh Edition of this work has been reset in new type
throughout. This has afforded the author an opportunity of making
some extensive revisions and of adding considerably to many of the
sections.
In the new matter will be found many practical hints, suggestions for
solving formula?, new methods of extracting cube roots and of finding
small powers and roots, a section on the determination of gauge
points, etc. Various new instruments are described and several ad-
ditional illustrations have been introduced.
On the vexed question of fixing the position of the decimal point, the
author feels that it is not possible, or at any rate, not desirable, to
dogmatise. In this matter the personal factor has to be reckoned with,
and while most practised users experience no difficulty in estimating the
magnitude of the result by inspection or rough calculation, others find
rules of considerable assistance. Buth methods have been discussed so
that the reader will be able to follow his own inclination in the matter.
The author tenders his thanks to the many who have evinced their
appreciation of his efforts to popularise the subject ; also for the many
kind hints and suggestions which he has received from time to time,
and with a continuance of which he trusts to be favoured in the
future. C. N. P.
Fallowfield, Manchester, May 1908.
177115
CONTENTS.
Introductory
The Mathematical Principle of the Slide Rule
Notation by Powers of 10
The Mechanical Principle of the Slide Rule
The Primitive Slide Rule
The Modern Slide Rule
The Notation of the Slide Rule
The Cursor or Runner ... ...
Multiplication ... ... ^,-
Division
The Use of the Upper Scales for Multiplication and Di
Reciprocals ...
Continued Multiplication and Division ...
Multiplication and Division with the Slide Inverted
Proportion ...
General Hints on the Elementary Uses of the Slide Ru
Squares and Square Roots
Cubes and Cube Roots
Miscellaneous Powers and Roots
Powers and Roots by Logarithms
Other Methods of Obtaining Powers and Roots
Combined Operations
Hints on Evaluating Expressions
Gauge Points
Examples in Technical Calculations
Trigonometrical Application
Slide Rules with Log-log Scales
Special Types of Slide Rules
Long Scale Slide Rules
Circular Calculators
Slide Rules for Special Calculations
Constructional Improvements in Slide Rule
The Accuracy of Slide Rule Results
Slide Rule Data Slips
6
8
9
10
12
14
17
19
24
26
27
28
30
31
36
37
40
45
45
47
49
52
56
74
84
92
96
101
10!)
110
111
113
^ OF THE
UNIVERSITY
OF
*£>UJFOB^
THE SLIDE RULE.
INTRODUCTORY.
THE slide rule may be defined as an instrument for mechani-
cally effecting calculations by logarithms. Those familiar
with logarithms and their use will recognise that the slide rule
provides what is in effect a concisely arranged table of logarithms,
together with a simple and convenient means for adding and
subtracting any selected values. Those, however, w T ho have no
acquaintance with logarithms will find that only an elementary
knowledge of the subject is necessary to enable them to make full
use of the slide rule. It is true that for simple slide-rule opera-
tions, as multiplication and division, a knowledge of logarithms is
unnecessary ; indeed, many who have no conscious understanding
of logarithms make good use of the' instrument. But this involves
a blind reliance upon rules without an appreciation of their origin
or limitations, and this, in turn, engenders a want of confidence in
the results of any but the simplest operations, and prevents the
fullest use being made of the instrument. For this reason a brief,
but probably sufficient resume of the principles of logarithmic
calculation will be given. Those desiring a more detailed ex-
planation are referred to the writer's "Logarithms for Beginners."
The slide rule enables various arithmetical, algebraical and
trigonometrical processes to be performed with ease and rapidity,
and with sufficient accuracy for most practical purposes. A grasp
of the simple fundamental principles which underlie its operation,
together with a little patient practice, are all that are necessary to
acquire facility in using the instrument, and few who have become
proficient in this system of calculating would willingly revert to
the laborious arithmetical processes.
5
THE SLIDE RULE:
THE MATHEMATICAL PRINCIPLE OF THE
SLIDE RULE.
Logarithms may be denned as a series of numbers in arithmetical
progression, as 0, 1, 2, 3, 4, etc., which bear a definite relationship
to another series of numbers in geometrical progression, as 1, 2, 4,
8, 16, etc. A more precise definition is : — The logarithm of a
number to any base, is the index of the power to which the base
must be raised to equal the given number. In the logarithms in
general use, known as common logarithms, and with which we are
alone concerned, 10 is the base selected. The general definition
may therefore be stated in the following modified form : — The
common logarithm of a number is the index of the poiver to which
10 must be raised to equal the given number. Applying this rule
to a simple case, as 100= 10 2 , we see that the base 10 must be
squared (i.e., raised to the 2nd power) in order to equal 100, the
number selected. Therefore, as 2 is the index of the power to
which 10 must be raised to equal 100, it follows from our definition
that 2 is the common logarithm of 100. Similarly the common
logarithm of 1000 will be 3, while proceeding in the opposite
direction the common log. of 10 must equal 1. Tabulating these
results and extending, we have : —
Numbers 10,000 1000 100 10 1
Logarithms 4 3 2 1
It will now be evident that for numbers
between 1 and 10 the logs, will be
10 „ 100
„ 100 „ 1000
„ 1000 „ 10,000
In other words, the logarithms of numbers between 1 and 10 will
be wholly fractional (i.e., decimal); the logs, of numbers between
10 and 100 will be 1 followed by a decimal quantity; the logs, of
numbers between 100 and 1000 will be 2 followed by a decimal
quantity, and so on. These decimal quantities for numbers from
1 to 10 (which are the logarithms of this particular series) are
as follows : —
Numbers ... 1 2 3 4 5 6 7 8 9 10
Logarithms 0-301 0-177 0-602 0*699 0*778 0-845 0-903 0*954 1-000
between
and 1
n
5J
1
2
3
■ ■
5} «
» 3
» 4
A PRACTICAL MANUAL 7
Combining the two tables, we can complete the logarithms. Thus
for 3 multiplied successively by 10, we have : —
Numbers... 3 30 300 3000 30,000
Logarithms 0*477 1*477 2-477 3*477 4*477 e C#
We see from this that for numbers having the same significant
figure (or figures), 3 in this case, the decimal part or mantissa of
the logarithm is the same, but that the integral part or character-
istic is always one less than the number of figures before the decimal
point.
For numbers less than 1 the same plan is followed. Thus
extending our first table downwards, we have : —
etc.
etc.
Numbers... 1 0*1 0*01 0*001 0*0001
Logarithms ...... -1 -2 -3 -4
so that for 3 divided successively by 10, we have : —
Numbers... 3 _0*3 _0'03 0-003 0/0003
Logarithms 0-477 1*477 2*477 3*477 4*477
Here again we see that with the same significant figures in the
numbers, the mantissa of the logarithm has always the same
(positive) value, but the characteristic is one more than the number
of 0's immediately following the decimal point, and is negative, as
indicated by the minus sign written over it. Only the decimal
parts of the logarithms of numbers between 1 and 10 are given
in the usual tables, for, as shown above, the logarithms of all ten-
fold multiples or submultiples of a number can be obtained at
once by modifying the characteristic in accordance with the rules
given.
An examination of the two rows of figures giving the logarithms
of numbers from 1 to 10 will reveal some striking peculiarities,
and at the same time serve to illustrate the principle of logarithmic
calculation. First, it will be noticed that the addition of any two
of the logarithms gives the logarithm of the product of these two
numbers. Thus, the addition of log. 2 and log. 4 = 0*301 + 0*602
= 0*903, and this is seen to be the logarithm of 8, that is, of 2 x4.
Conversely, the difference of the logarithms of two numbers gives
the logarithm of the quotient resulting from the division of these
two numbers. Thus, log. 8 -log. 2 = 0*903 -0*301 = 0*602, which is
the log. of 4, or of 8 + 2.
8 THE SLIDE RULE:
One other important point is to be noted. If the logarithm of
any number is multiplied by 2, 3, or any other quantity, whole or
fractional, the result is the logarithm of the original number,
raised to the 2nd, 3rd, or other power respectively. Thus, multi-
plying the log. of 3 by 2, we obtain 0*477x2 = 0954, and this is
seen to be the log. of 9, that is, of 3 raised to the 2nd power, or 3
squared. Again, log. 2 multiplied by 3 = 0*903 — that is, the log. of
8, or of 2 raised to the 3rd power, or 2 cubed. Conversely, dividing
the logarithm of any original number by any number n, we obtain
the logarithm of the nth. root of the original number. Thus, log.
8-r3 = 0*903-r3^0*301, and is therefore equal to log. 2 or to the
log. of the cube root of 8.
Only simple logs, have been taken in these examples, but the
student will understand that the same reasoning applies, whatever
the number. Thus for 20 3 we prefix the characteristic (1 in this
case) to log. 2, giving 1*301. Multiplying by 3, we have 3*903 as
the resulting logarithm, and as its characteristic is 3, we know
that it corresponds to the number 8000. Hence 20 3 =8000.
In this brief explanation is included all that need now be said
with regard to the properties of logarithms. The main facts to
be borne clearly in mind are : — (1.) That to find the product of
two numbers, the logarithms of the numbers are to be added
together, the result being the logarithm of the product required,
the value of which can then be determined. (2.) That in finding
the quotient resulting from the division of one number by another,
the difference of the logarithms of the numbers gives the logarithm
of the quotient, from which the value of the latter can be ascer-
tained. (3.) That to find the result of raising a number to the nth
power, we multiply the logarithm of the number by n, thus obtain-
ing the logarithm, and hence the value, of the desired result. And
(4.) That to find the nth root of a number, we divide the logarithm
of the number by n, this giving the logarithm of the result, from
which its value may be determined.
NOTATION BY POWERS OF 10.
A convenient method of representing an arithmetical quantity is
to split it up into two factors, of which the first is the original
number, with the decimal point moved so as to immediately follow
the first significant figure, and the second, 10" where n is the
A PRACTICAL MANUAL
number of places the decimal point has been moved, this index
being positive for numbers greater than 1, and negative for numbers
Less than 1.* In this system, therefore, we regard 3,610,000 as
3*61 x l,000,000,and write it as 3'61 x 10 6 . Similarly 361 =3*61 x 10 2 ;
0*0361
=^) =3-
100/
•61 x 10- 2 ; 0*0000361 =3*61 x 10" 5 , etc. To re-
store a number to its original form, we have only to move the
decimal point through the number of places indicated by the
index, moving to the right if the index is positive and to the
left (prefixing 0's) if negative. This method, which should be
cultivated for ordinary arithmetical work, is substantially that
followed in calculating by the slide rule. Thus with the slide rule
the multiplication of 63,200 by 0*0035 virtually resolves itself into
6'32 x 10 4 x 3*5 x 10" 3 or 6*32 x 35 x 10 4 - 3 = 22*12 x HP « 221*2, It will
be seen later, however, that the result can be arrived at by a more
direct, if less systematic, method of working.
THE MECHANICAL PRINCIPLE OF THE SLIDE RULE.
The mechanical principle involved in the slide rule is of a very
simple character. In Fig. 1, A and B represent two rules divided
into 10 equal parts, the division lines being numbered consecutively
£
0-7
^
A
1
2
3
4
5
6
' 7
8
10
16
— 0-3
B
1
1
2
3
4
6
6
7
8
9
10
1
O-*—
»l
-
Fig. 1.
as shown. If the rule B is moved to the right until on B is
opposite 3 on A, it is seen that any number on A is equal to the
coinciding number on B, plus 3. Thus opposite 4 on B is 7 on A.
The reason is obvious. By moving B to the right, we add to a
length 0-3, another length 0-4, the result read off on A being 7.
Evidently, the same result would have been obtained if a length
0-4 had been added, by means of a pair of dividers, to the length
0-3 on the scale A. By means of the slide B, however, the addition
is more readily effected, and, what is of much greater importance,
* It will be recognised that n is the characteristic of the logarithm of
the original number.
10
THE SLIDE RULE:
the result of adding 3 to any one of the numbers within range, on
the lower scale, is immediately seen by reading the adjacent
number on A.
Of course, subtraction can be quite as readily performed.
Thus, to subtract 4 from 7, we require to deduct from 0-7 on the
A scale, a length 0-4 on B. We do this by placing 4 on B under
7 on A, when over on B we find 3, on A. It is here evident that
the difference of any pair of coinciding numbers on the scales is
constantly equal to 3.
An important modification results if the slide-scale B is in-
verted as in Fig. 2. In this case, to find the sum of 4 and 3 we
require to place the 4 of the A scale to 3 on the B scale, and the
result is read on A over on B. Here it will be noted, the sum
of any pair of coinciding numbers on the scales is constant and
A
1
1
3
4
6
a
7
8
9
m
3
o»
6
8
I
9
7
1
e
I
1
Fig. 2.
equal to 7. This case, therefore, resembles that of the immediately
preceding one, except that the sum, instead of the difference, of any
pair of coinciding numbers is constant.
To find the difference of two factors, the converse operation is
necessary. Thus, to subtract 4 from 7, on B is placed opposite 7
on A, and over 4 on B is found 3 on A.
From these examples it will be seen that with the slide in
the methods of operation are the reverse of those used when the
slide is in its normal position.
It will be understood that although we have only considered
the primary divisions of the scales, the remarks apply equally to
any subdivisions into which the primary spaces of the scales might
be divided. Further, we note that the length of scale taken to
represent a unit is quite arbitrary.
THE PEIMITIVE SLIDE BULE.
The application of the foregoing principles to the slide rale can
be shown most conveniently by describing the construction of a
simple form of slide rule : — Take a strip of card about 11 in. long
and 2in. wide ; draw a line down the centre of its width, and
A PRACTICAL MANUAL
11
mark off two points, lOin. apart. Draw cross lines at these points
and figure them 1 and 10 on each side, as in Fig. 3. Next mark off
lengths of 3'01, 477, 6'02, 6*99, 7'78, 8'45, 9'03 and 9*54 inches,
from the line marked 1. Draw cross lines as before, and figure
these lines, 2, 3, 4, 5, 6, 7, 8 and 9. To fill in the intermediate
divisions of the scale, take the logs, of 1*1, 1*2, 1*3, etc. (from a
table), multiply each by 10, and thus obtain the distances from 1,
.
„
1 3 01' >,
I 4 5 6 7 8 9|
1
! C
:
4 6 6
7 8 1
Fig. 3.
at which the several subdivisions are to be placed. Mark these
1*2, 1*3, 1*4, etc., and complete the scale, making the interpolated
division marks shorter to facilitate reading, as with an ordinary
measuring rule. Cutting the card cleanly down the centre line,
we have the essentials of the slide rule.
The fundamental principle of the slide rule is now evident : —
Each scale is graduated in such a manner that the distance of any
number from 1 is proportional to the logarithm of that number.
We know that to find the product of 2x3 by logarithms, we
add 0-301, or log. 2, to 0'477, the log. of 3, obtaining 0*778, or log.
6. With our primitive slide rule we place 1 on the lower scale to
Fig. 4-.
3'Olin. (which we have marked 2) on the upper scale (Fig. 4). Then
over 4"77in. on the lower scale (which we marked 3), we have 7'78in.
(which we marked 6) on the upper scale. Conversely, to divide
6 by 3, we place 3 on the lower scale in agreement with 6 on the
upper, and over 1 on the lower scale read 2 on the upper scale.
This method of adding and subtracting scale lengths will be seen
to be identical with that used in the simple case shown in Fig. 1.
12
THE SLIDE RULE:
THE MODEEN SLIDE BULE.
The modern form of slide rule, variously
styled the Gravet, the Tavernier- Gravet,
and the Mannheim rule, is frequently made
of boxwood, but all the leading instrument
makers now supply rules made of boxwood
or mahogany, and faced with celluloid, the
white surface of which brings out the
graduations much more distinctly than lines
engraved on a boxwood surface. The cellu-
loid facings should not be polished, as a dull
surface is much less fatiguing to the eyes.
The most generally used, and on the whole
the most convenient size of rule, is about
10 Jin. long, ljin. wide, and about gin. thick ;
but 5in., 8in., 15in., 20in., 24in. and 40in.
rules are also made. In the centre of the stock
of the rule a movable slip is fitted, wljich
constitutes the slide, and corresponds to the
lower of the two rules of our rudiment? ry
examples.
From Fig. 5, which is a representation
of the face of a Gravet or Mannheim slide
rule, it will be seen that four series of loga-
rithmic graduations or scale-lines are em-
ployed, the upper and lower being engraved
on the stock or body of the rule, while the
other two are engraved upon the slide. The
two upper sets of graduations are exactly
alike in every particular, and the Lower
sets are also similar. It is usual to identify
the two upper scale-lines by the letters A
and B, and the two lower by the letters C
and I >, as indicated in the figure at the
left-hand exi remit ies of the scales.
Referring to the scales C and I \ these
will each be seen to be a development
of the element. -nv scales of Fig. 3, but
A PRACTICAL MANUAL 13
in this case each principal space is subdivided, more or
less minutely. The principle, however, is exactly the same,
so that by moving the slide (carrying scale C), multiplication
and division can be mechanically performed in the manner
described.
The upper scale-line A consists of two exactly similar scales,
placed end to end, the first lying between II and Ic, and the
second between Ic and Ir. The first of these scales w r ill be desig-
nated the left-hand A scale, and the second the right-hand A scale.
Similarly the coinciding scales on the slide are the left-hand B
scale and the right-hand B scale. Each of these four scales is
divided (as finely as convenient) as in the case of the C and D
scales, but, of course, they are exactly one half the length of the
latter.
The two end graduations of both the C and D scales are known
as the left- and right-hand indices of these scales. Sometimes they
are figured 1 and 10 respectively ; sometimes both are marked 1.
Similarly II and Ir are the left- and right-hand indices of the A
and B lines, while Ic is the centre index of these scales. Other
division lines usually found on the face of the rule are one on the
left-hand A and B scales, indicating the ratio of the circumference
of a circle to its diameter, tt= 3*1416 ; and a line on the right-hand
B scale marking the position of -=0'7854, used in calculating
the areas of circles. Reference will be made hereafter to the
scales on the under-side of the slide, and we need now only add
that one of the edges of the rule, usually bevelled, is generally
graduated in millimetres, while the other edge has engraved on it
a scale of inches divided into eighths or tenths. On the bottom
face inside the groove of the rule either one or the other of these
scales is continued in such a manner that by drawing the slide out
to the right and using the scale inside the rule, in conjunction
with the corresponding scale on the edge, it is possible to measure
20 inches in the one case, or nearly 500 millimetres in the other.
On the back of the rule there is usually a collection of data, for
which the slips given at the end of this work may often be sub-
stituted with advantage.
14 THE SLIDE RULE:
THE NOTATION OF THE SLIDE RULE.
Hitherto our attention has been confined to a consideration of
the primary divisions of the scales. The same principle of gradua-
tion is, however, used throughout ; and after what has been said,
this part of the subject need not be further enlarged upon. Some
explanation of the method of reading the scales is necessary, as
facility in using the instrument depends in a very great measure
upon the dexterity of the operator in assigning the correct value
to each division on the rule. By reference to Fig. 5, it will be
seen that each of the primary spacings in the several scales is
invariably subdivided into ten ; but since the lengths of the
successive primary divisions rapidly diminish, it is impossible to
subdivide each main space into the same number of parts that
the space 1-2 can be subdivided. This variable spacing of the
scales is at first confusing to the student, but with a little practice
the difficulty is soon overcome.
With the C or D scale, it will be noticed that the length of the
interval 1-2 is sufficient to allow each of the 10 subdivisions to be
again divided into 10 parts, so that the whole interval 1-2 is
divided into 100. The shorter main space 2-3, and the still shorter
one 3-4, only allow of the 10 subdivisions of each being divided
into live parts. Each of these main spaces is therefore divided
into 50 parts. For the remainder of the scale each of the 10
subdivisions of each main space is divided into two parts only ; so
that from the main division 4 to the end of the scale the primary
spaces are divided into 20 parts only.
In the upper scales A or B, it will be found that — as the space
1-2 is of only half the length of the corresponding space on C or 1 >
— the 10 subdivisions of this interval are divided into five puts
only. Similarly each of the 10 subdivisions of the intervals 2-3,
3-4, and 4-5 are further divided into two parts only, while
for the remainder of the scale only the 10 subdivisions are
possible, owing to the rapidly diminishing lengths of the primary
spacings.
The values actually given on the rule run from 1 to L0 on
the lower scales and from 1 to 100 on the upper scales, and, as
explained on page 9, all factors are brought within these ranges
of values by multiplying or dividing them by powers of 10, By
following this plan, we virtually regard each factor as merely a
A PRACTICAL MANUAL 15
series of significant figures, and make the necessary modification
due to the "powers of 10 ; ' when fixing the position of the decimal
point in the answer.
Many, however, find it convenient in practice to regard the
values on the rule as multiplied or divided by such powers of 10
as may be necessary to suit the factors entering into the calcula-
tion. If this plan is adopted, the values given to each graduation
of the scales will depend on that given to the left index figure (1)
of the lower scales, this being any multiple or submultiple of 10.
Thus II on the D scale may be regarded as 1, 10, 100, 1000, etc., or
as 0'1, 001, 0*001, 00001, etc. ; but once the initial value is assigned
to the index, the ratio of value must be maintained throughout the
whole scale. For example, if 1 on C is taken to represent 10, the
main divisions 2, 3, 4, etc., will be read as 20, 30, 40, etc. On
the other hand, if the fourth main division is read as 0004, then
the left index figure of the scale will be read as 0*001. The figured
subdivisions of the main space 1-2 are to be read as 11, 12, 13, 14,
15, 16, 17, 18 and 19 — if the index represents 10, — and as corre-
sponding multiples for any other value of the index.
Independently considered, these remarks apply equally to the
A or B scale, but in this case the notation is continued through
the second half of the scale, the figures of which are to be read as
tenfold values of the corresponding figures in the first half of the
scale.
The reading of the intermediate divisions will, of course, be
determined by the values assigned to the main divisions. Thus, if
II on D is read as 1, then each of the smallest subdivisions of the
space 1-2 will be read as 0*01, and each of the smallest subdivisions
of the spaces 2-3 or 3-4 as 0"02, while for the remainder of the
scale the smallest subdivisions are read as 0*05. In the A or B
scale the subdivisions of the space 1-2 of the first half of the scale
are (if Il = 1) read as 0'02, 0*04, etc. ; for the divisions 2-3, 3-4, and
4-5, the smallest intervals are read as 0*05 of the primary spaces,
and from 5 to the centre index of the scale the divisions represent
0'1 of each main interval. Passing the centre index, which is now
read as 10, the smallest subdivisions immediately following are
read 10'2, 10'4, etc., until 20*0 is reached ; then we read 20*5, 21*0,
21*5, 22*0, etc., until the figured main division 5 is reached. The
remainder of the scale is read 51, 52, 53, etc., up to 100, the right-
hand index.
16 THE SLIDE RULE*.
Further subdivision of any of the spaces of the rule can be
effected by the eye, and after a little practice the operator will
become quite expert in estimating any intermediate value. It
affords good practice to set 1 on C to 1*04, 1*09, etc. on I), and to
read the values on D, under 4, 6, 8, etc. on C. As the exact
results are easily calculated mentally, the student, by this means,
will receive better instruction in estimating intermediate results
than can be given by any diagram.
Some rules will be found figured as shown in Fig. 5 ; in others,
the right-hand upper scales are marked 10, 20, 30, etc. Again,
others are marked decimally, the lower scales and the left-hand
upper scales being figured 1, 1*1, 1*2, 1*3 2*5, etc. The
latter form has advantages from the point of view of the
beginner.
The method of reading the A and B scales, just given, applies
only when these scales are regarded as altogether independent of
the lower pair of scales C and D. Some operators prefer to use
the A and B scales, and some the C and D scales, for the ordinary
operations of proportion, multiplication, and division. Each
method has its advantages, as will be shown, but in the more
complex calculations, as involution and evolution, etc., the relation
of the upper scales to the lower scales becomes a very important
factor.
The distance 1-10 on the upper scales is one-half of the distance
1-10 on the lower scales. Hence any distance from 1, taken on
the upper scales, represents twice the logarithm which the same
distance represents on the lower scales. In other words, the length
which represents log. N on D, would represent 2 log. N on A ;
and, conversely, the length which represents log. N on A, would
represent * . on D.
Now we have seen (page 8) that multiplying the log. of a
number by 2 gives the log. of the square of the number. Hence,
above any number on D we find its square on A, or, conversely,
below any number on A, we find its square root on D. Tims,
above 2 we find 4 ; under 49, we find 7 and so on. Obviously the
same relation exists between the B and C scales.
A PRACTICAL MANUAL 17
THE CUKSOR OE RUNNER
All modern slide rules are now fitted with a cursor or nmner,
which usually consists of a light metal frame moving under spring
control in grooves in the edges of the stock of the rule. This
frame carries a piece of glass, mica or transparent celluloid, about
lin. square, across the centre of which a fine reference line is
drawn exactly at right angles to the line of scales. To "set the
cursor" to any value on the scales of the rule, the frame is taken
between the thumb and forefinger and adjusted in position until
the line falls exactly upon the graduation, or upon an estimated
value, between a pair of graduations, as the case may be. Having
fixed one number in this way, another value on either of the scales
on the slide may be similarly adjusted in reference to the cursor
line. The cursor will be found very convenient in making such
settings, especially when either or both of the numbers are located
by eye estimation. It also finds a very important use in referring
the readings of the upper scale to those of the lower, or vice versa,
while as an aid in continued multiplication and division and com-
plex calculations generally, its value is inestimable.
Multiple Line Cursors. — Cursors can be obtained with two lines,
the distance between them being that between 7*854 and 10 on
the A scale. The use of this cursor is explained on page 57.
Another multiple line cursor has short lines engraved on it, corre-
sponding to the main graduations from 95 to 105 on the respective
scales. This is useful for adding or deducting small percentages.
The Broken Line Cursor. — To facilitate setting, broken line
cursors are made, in which the hair-line is not continued across
the scales, but has two gaps, as shown in Fig. 6.
The Pointed Cursor has an index or pointer, extending over the
bevelled edge of the rule, on which is a scale of inches. It is
useful for summing the lengths of the ordinates of indicator
diagrams, and also for plotting lengths representing the logarithms
of numbers, sometimes required in graphic calculations.
The Goulding Cursor. — It has been pointed out that in order to
obtain the third or fourth figure of a reading on the 10-in. slide
rule, it is frequently necessary to depend upon the operator's
ability to mentally subdivide the space within which the reading
falls. This subdivision can be mechanically effected by the aid of
the Goulding Cursor (Fig. 7), which consists of a frame fitting
18
THE SLIDE RULE
into the usual grooves in the rule, and carrying a metal plate faced
with celluloid, upon which is engraved a triangular scale ABC.
The portion carrying the chisel edges E is not fixed to the cursor
proper, but slides on the latter, so that the index marks on the
projecting prongs can be moved slightly along the scales, of the
rule, this movement being effected by the short end of the bent
lever F working in the slot as shown. D is a pointer which can
be moved along F under spring control. As illustrating the
method of use, we will assume that 1 on C is placed to 155 on D,
and that we require to read the value on D under 27 on C. This
&3
<*
i
1 3
: ;r^
Fig. 8.
Fig. 7.
Fig. 9.
is seen to lie between 4150 and 4200, so setting the pointer. D to
the line B — always the first operation — we move the whole
along the rule until the index line on the lower prong agrees with
4200. We then move F across the scale until the index line agrees
with 4100, set the pointer 1) to the line A C, and move the lever
back until the index line agrees with 27 on the slide. It will then
be found that the pointer D gives 85 on A B as the value of the
supplementary figures, and hence the complete reading is 4185.
Magnifying Cursors are of assistance in reading the scales, and
in. a good and direct light are very helpful. In one form an
ordinary lens is carried by two light arms hinged to the Upper
and lower edges of the cursor, so that it can be folded down to the
face of the rule when not in use. A more compact form, shown in
A PRACTICAL MANtJAt
10
Fig. 8, consists of a strip of plano-convex glass, on the under side
of which is the hair-line. In a cursor made by Nestler of Lahr,
the plano-convex strip is fixed on the ordinary cursor. The
magnifying power is about 2, so that a 5in. rule, having the same
number of graduations as a lOin. rule, can be read with equal
facility, by the aid of /this cursor.
The Digit-registering Cursor, supplied by Mr A. W, Faber,
London, and shown in Fig. 9, has a semicircular scale running
from at the centre upward to — 6 and downward to + 6. A
small linger enables the operator to register the number of digits
to be added or subtracted at the end of a lengthy operation, as
explained at page 28.
MULTIPLICATION.
In the preliminary notes it was shown that by mechanically
adding two lengths representing the logarithms of two numbers, we
can obtain the product of these numbers ; while by subtracting one
n
,
2
[
3
4 5
[ ]
G
|
7 10
i I l [
1
3
4
5
6
7 8 9 10 1
Fig. 10.
log. -length from another, the number represented by the latter is
divided by the number represented by the former. Hence, using
the C and D scales, we have the
Rule for Multiplication.— Set the index of the C scale to one of the
factors on D, and %mder the ether factor on C,Jind the product on D.
Thus, to find the product of 2 x 4, the slide is moved to the right
until the left index (1) of C is brought over 2 on D, when under the
other factor (4) on C, is found the required product (8) on D.
Following along the slide, to the right, we find that beyond 5 on C
(giving 10 on D), W3 have no scale below the projecting slide (Fig.
10). If we imagine the D scale prolonged to the right, we should
have a repetition of the earlier portion, but, as with the two parts
20 THE SLIDE RULE:
of the A scales, the repeated portion would be of tenfold value, and
10 on C would agree with 20 on the prolonged D scale. We turn
this fact to account by moving the slide to the left until 10 on C
agrees with 2 on T), and we can then read off such results as
2x6 = 12 ; 2x8 = 16, etc., remembering that as the scale is now of
tenfold value, there will be two figures in the result. Hence, for
those who prefer rules, we have the
Rule for the Number of Digits in a Product. — If the pro-
duct is read with the slide projecting to the left, add the number
of the digits in the two factors ; if read with the slide to the
right, deduct 1 from this sum.
Ex.-25x 70 = 1750.
The product is found with the slide projecting to the left, so
the number of digits in the product = 2 +2 = 4.
Ex.— 3*6x25 = 90.
The slide projects to the right, and the number of digits in the
product is therefore 1 + 2-1 = 2.
Ex.— 0-025x0-7 = 0-0175.
The product is obtained with the slide projecting to the left,
and the number of digits is therefore -1 + 0= - 1.
Ex.— 0-000184x0-005 = 0-00000092.
The sum of the number of digits in the two factors = - 3 + ( - 2)
= - 5, but as the slide projects to the right, the number of digits
will be -5 -1= -6.
From the last two examples it will be seen that when the first
significant figure of a decimal factor does not immediately follow
the decimal point, the minus sign is to be prefixed to the number
of digits, counting as many digits minus as there are 0's following
the decimal point. Thus, 0'03 has - 1 digit, 0*0035 has -2 digits,
and so on. Some little care is necessary to ensure these minus
values being correctly taken into account in determining the
number of digits in the answer. For this reason many prefer to
treat decimal factors as whole numbers, and to locate- the decimal
point according to the usual rules for the multiplication of decimals.
Thus, in the last example we take 184 x 5 = 920, but as by the usual
rule the product must contain 6 + 3 = 9 decimal places, we prefix
six cyphers, obtaining 0*00000092. When both factors consist of
integers as well as decimals, the number of digits in the product,
and therefore the position of the decimal point, will be determined
by the usual rule for whole numbers.
A PRACTICAL MANUAL 21
Another method of determining the number of digits in a
product deserves mention, which, not being dependent upon the
position of the slide, is applicable to all calculating instruments.
General Kule for Number of Digits in a Product. — When
the first significant figure in the product is smaller than in either of
the factors, the number of digits in the product is equal to the sum of
the digits in the two factors. When the contrary is the case, the
number of digits is 1 less than the sum of the digits in the two
factors. When the first figures are the sa?ne, those following must be
compared.
Estimation of the Figures in a Product.— We have given rules for
those who prefer to decide the number of figures by this means,
but experience will show that to make the best use of the instru-
ment, the result, as read on the rule, should be regarded merely
as the significant figures of the answer, the position of the decimal
point, if not obvious, being decided by a very rough mental calcu-
lation. In very many instances, the magnitude of the result will
be evident from the conditions of the problem — e.g., whether the
answer should be 0'3in., 3in., or 30in. ; or 10 tons, 0*1 ton, 100
tons, etc. In those cases where the magnitude of the answer can-
not be estimated, and the factors contain many figures, or have a
number of O's following the decimal point, the use of notation by
powers of 10 (page 8) is of considerable assistance ; but more
usually it will be found that a very rough calculation will settle the
point with comparatively little trouble. Considerable practice is
needed to work rapidly and with certainty, when using rules.
Moreover, the experience thus acquired is confined to slide-rule
work. The same time spent in practising the "rough approxi-
mation" method will enable reliable results to be obtained
rapidly, with the advantage that the method is applicable to
calculations generally. However, the choice of methods is a
matter of personal preference. Both methods will be given, but
whichever plan is followed, the student is strongly advised to
cultivate the habit of forming an idea of the magnitude of the
result.
Ex.— 33*6x236 = 7930.
Setting 1 on C to 33*6 on D, we read under 236 on D and find 793
on D, as the significant figures of the answer. A rough calculation,
as 30x200 = 6000, indicates that the result will consist of 4 figures,
and is therefore to be read as 7930.
22 THE SLibM rule:
Ex.— 17,300 x 3780 = 65,400,000.
By factorizing with powers of 10
1-73 x 10 4 x 3-78 x 10 3 = 1'73 x 3'78 x 10 7 .
Setting 1 on C to 1'73 on D, we read, under 3-78 on C, the result
of the simple multiplication, as 6 "54. Multiplying by 10 7 moves, the
decimal point 7 places to the right, and the answer is 65,400,000.
If it is required to find a series of products of which one of the
factors is constant, set 1 on C to the constant factor on D and read
the several products on D, under the respective variable factors.
If the factors are required which will give a constant 'product
(really a case of division), set the cursor to the constant product on
D. Then obviously, as the slide is moved along, any pair of factors
found simultaneously under the cursor line on C, and on D under
index of C, will give the product. A better method of working
will be explained when we deal with the inversion of the slide.
It is sometimes useful to remember that although we usually
set the slide to the rule, we can obtain the result equally well In-
setting the rule to the slide. Thus, bringing 1 (or 10) on D to 2
on C, we find on C, over any other factor, n on D, the product of
2 x n. But note that the slide and rule have now changed places,
and if we use rules for the number of digits in the result, we
must now deduct 1 from the sum of the digits in the factors,
when the rule projects to the right of the slide.
With the ordinary lOin. rule it will be found in general that
the extent to which the C and D scales are subdivided is such as
to enable not more than three figures in either factor being dealt
with. For the same reason it is impossible to directly read more
than the first three figures of any product, although it is often
possible— by mentally dividing the smallest space involved in the
reading — to correctly determine the fourth figure of a product.
Necessarily this method is only reliable when used in the earlier
parts of the C and D scales. However, the last numeral of a
three-figure, and in some cases the last of a four-figure, product
can be readily ascertained by an inspection of the factors.
Ex.— 19x27 = 513. Placing the L.H. index of C to 19 on I),
we find opposite 27 on C, the product, which lies between 510 and
516. A glance at the factors, however, is sufficient to decide that
the third figure must be 3, since the product of \) and 7 is (IH. and
the last figure of this product must be the last figure in the answer,
Ex.— 79x91 = 7189.
UNIVERSITY J
A PRACTICAL MANUAL 23
In this case the division line 91 on C indicates on D that the
answer lies between 7180 and 7190. As the last figure must be 9,
it is at once inferred that the last two figures are 89.
When there are more than three figures in either or both of the
factors, the fourth and following figures to the right must be
neglected. It is well to note, however, that if the first neglected
figure is 5, or greater than 5, it will generally be advisable to
increase by 1 the third figure of the factor employed. Generally
it will suffice to make this increase in one of the two factors only,
but it is obvious that in some cases greater accuracy will be
obtained by increasing both factors in this way.
Continued Multiplication. — To find the product of more
than two factors, we make use of the cursor to mark the position
of successive products (the value of which does not concern us) as
the several factors are taken into the calculation. Setting the index
of C to the 1st factor on D, we bring the line of the cursor to the
2nd factor on C, then the index of C to the cursor, the cursor to
the 3rd factor, index of C to cursor, and so on, reading the final pro-
duct on D under the last factor on C. (Note that the 1st factor and
the result are read on D ; all intermediate readings are taken on C.)
If the rule for the number of digits in a product is used, it is
necessary to note the number of times multiplication is effected
with the slide projecting to the right. This number, deducted
from the sum of the digits of the several factors, gives the number
of digits in the product. Ingenious devices have been adopted to
record the number of times the slide projects to the right, but
some of these are very inconvenient. The author's method is to
record each time the slide so projects, "by a minus mark, thus - .
These can be noted down in any convenient manner, and the sum
of the marks so obtained deducted from the sum of the digits in
the several factors, gives the number of digits in the product as
before explained.
Ex.— 42 x 71 x 1-5 x 0-32 x 121 = 173,200.
The product given, which is that read on the rule, is obtained
as follows :— Set r.h. index of C to 42 on D, and bring the cursor
to 71 on C. Next bring the l.h. index of C to the cursor, and the
latter to 1 *5 on C. This multiplication is effected with the slide
to the right, and a memorandum of this fact is kept by making a
mark - . Bring the r.h. index of C to the cursor and the latter to
0'32 on C. Then set the l.h. incfex of Q to the cursor and read
24 THE SLIDE RULE:
the result, 1732, on D under 121 on C, while as the slide again
projects to the right, a second - memo-mark is recorded. There
are 2 + 2 + 1+0 + 3 = 8 digits in the factors, and as there were 2 -
marks recorded during the operation, there will be 8-2 = 6 digits
in the product, which will therefore read 173,200. The true
product is 173,194'56.
For a very rough evaluation of the result, we note that 1*5 x 0*3
is about 0'5 ; hence, as a clue to the number of figures we have
40 x 70 x 60 = 3000 x 60 = 180,000.
DIVISION.
The instructions for multiplication having been given in some detail,
a full discussion of the inverse process of division will be unnecessary.
.Rule for Division. — Place the divisor on (7, opposite the dividend
on Dj and read the quotient on D under the index of C.
Ex.— 225 + 18 = 12-5.
Bringing 18 on C to 225 on D, we find 12*5 under the l.h.
index of C.
As in multiplication, the factors are treated as whole numbers,
and the position of the decimal point afterwards decided according
to the following rule, which, as will be seen, is the reverse of that
for multiplication : —
The Number of Digits in a Quotient. — If the quotient is read
with the slide projecting to the left, subtract the number of digits in
the divisor from those in the dividend; but if read with the slide to
the right, add 1 to the difference of the number of digits.
In the above example the quotient is read off with the slide to
the right, so the number of digits in the answer = 3 - 2 + 1 = 2.
Ex.— 0000221 +0-017 = 0'013.
Here the number of digits in the dividend is -3, And in the
divisor -1. The difference is- 2; but as the result is obtained
with the slide to the right, this result must be increased by 1, so
that the number of digits in the quotient is - 2 + 1 = - 1, giving the
answer as 0*013.
If preferred, the result can be obtained in the manner referred
to when considering the multiplication of decimals. Thus, treating
the above as whole numbers, we find that the result of dividing
221 by 17 = 13, since the difference in the number of digits in the
factors, which is 1, is, owing to \,he position of the slide, increased
by 1, giving 2 as the number of digits in the answer. Then by the
A PRACTICAL MANUAL 25
rules for the division of decimals we know that the number of
decimal places in the quotient is equal to 6-3 = 3, showing that a
cypher is to be prefixed to the result read on the rule.
As in multiplication, so in division, we have a
General Eule for Number of Digits in a Quotient.— When
the first significant figure in the divisor is greater than that in the
dividend, the number of digits in the quotient is found by subtracting
the digits in the divisor from those in the dividend. When the contrary
is the case, 1 is to be added to this difference. When the first figures
are the same, those following must be compared.
Estimation of the Figures in a Quotient. — The method of
roughly estimating the number of figures in a quotient needs little
explanation.
Ex.— 3*95 -r 5340 = 0-00074.
Setting 534 on C to 3*95 on D we read under the (r.h.) index of
C, the significant figures on D, which are 74. Then 3 9-^5 is about 0*8
and '8-^1000 gives 0*0008 as a rough estimate.
Ex. —0 -00000285 -f -000197 = -01446.
Regarding this as 2-85 x lO-^-fl-97 x 10" 4 we divide 2-85 by 1*97
and obtain 1*446. Dividing the powers of 10 we have 10 ~ 6 -*- 10 "" 4 =
10 " 2 , so the decimal point is to be moved two places to the left and the
answer is read as 0*01446.
Another method of dividing deserves mention as of special
service when dividing a number of quantities by a constant divisor: —
Set the index of C to the divisor on D and over any dividend on D,
read the quotient on C.
For the division of a constant dividend by a variable divisor,
set the cursor to the dividend on D and bring the divisor on C
successively to the cursor, reading the corresponding quotients on
D under the index of C. Another method which avoids moving
the slide is explained in the section on " Multiplication and Division
with the Slide Inverted."
Continued Division, if we can so call such an expression as
— =0-0688,
785 x 0-00021 x 4-3x64-4 '
may be worked by repeating as follows : — Set 7'85 on C to 3*14 on
D, bring cursor to index of C, 2*1 on C to cursor, cursor to index,
4'3 to cursor, cursor to index, 6'44 to cursor, and under index of C
read 688 on D as the significant figures of the answer.
For the number of figures in the result, we deduct the sum of
the number of dibits in the several factors and add 1 for each
26 THE SLIDE RULE:
time the slide projects to the right, which in this case occurs once.
There are 3 + (-3) + l + 2 = 3 denominator digits, 1 numerator digit,
and 1 is to be added to the difference. Therefore there are
1-3 + 1= —1 digits in the answer, which is therefore 00688. The
foregoing method of working may confuse the beginner, who is apt
to fall into the process of continued multiplication. For this
reason, until familiarity with combined methods has been acquired,
the product of the several denominators should be first found by
the continued multiplication process, and the figures in this product
determined. Then divide the numerator by this product to obtain
the result.
As the denominator product will be read on D, we may avoid
resetting the slide by bringing the numerator on C to this product
and reading the result on C over the index of D. The slide and
rule have here changed places ; hence if rules are followed for the
number of figures in the result, 1 must be added to the difference
of digits, when the rule projects to the right of the slide.
The author's method of recording the number of times division
is performed with the slide to the right is by vertical memorandum
marks, thus I. The full significance of these memo-marks will
appear in the following section.
For a rough calculation to fix the decimal point, in this example
we move the decimal points in the factors, obtaining
3 _ 3
0-8x2x4x6 - 40~ ' ' 5 -
THE USE OF THE UPPER SCALES FOR
MULTIPLICATION AND DIVISION.
Many prefer to use the upper scales A and B, in preference to C
and D. The disadvantage is that as the scales are only one-half
the length of C or D, the graduation does not permit of the same
degree of accuracy being obtained as when working with the lower
scales. But the result can always be read directly from the rule
without ever having to change the position of the slide after it has
been initially set. Hence, it obviates the uncertainty as to the
direction in which the slide is to be moved in making a setting.
When the A and B scales are employed, it is understood that
the left-hand pair of scales are to be used in the same manner as
C and D, and so far the rules relating to the latter are entirely
applicable. But in this case the slide is always moved to the
A PRACTICAL MANUAL 27
right, so that in multiplication the product is found either upon
the left or right scales of A. If it is found on the left A scale,
the rule for the number of digits in the product is found as for
the C and D scales, and is equal to the sum of the digits in the two
factors, minus 1 ; but if found on the right-hand A scale, the
number of digits in the product is equal to the sum of the digits
in the two factors.
In division, similar modifications are necessary. If when
moving the slide to the right the division can be completely
effected by using the l.h. scale of A, the quotient (read on A above
the l.h. of index B) has a number of digits equal to the number in
the dividend, less the number in the divisor, plus 1. But if the
division necessitates the use of both the A scales, the number of
digits in the quotient equals the number in the dividend, less the
number in the divisor.
EECIPEOCALS.
A srEciAL case of division to be considered is the determination of
the reciprocal of a number n, or -. Following the ordinary rule for
division, it is evident that setting n on C to 1 on D, gives * on D
n
under 1 on C It is more important to observe that by inverting
the operation— setting 1 (or 10; on C to n on D — we can read - on
n
C over 1 (or 10) on D. Hence whenever a result is read on D
under an index of C, we can also read its reciprocal on C over
whichever index of D is available.
The Numher of Digits in a Reciprocal is obvious when ft = 10
100, or any power (p) of 10. Thus l =01 ; — = 0"01 ; _L = 1
preceded by p - 1 cyphers. For all other cases we have the rule :
Subtract from 1 the number of digits in the number,
Ex.— JL = 0-00295.
339
There are 3 digits in the number ; hence, there are 1-3= -2
digits in the answer.
Ex.— 1 = 64,100.
0-0000156
There are - 4 digits in the number ; hence, there are 1 - ( - 4) = 5
digits in the result.
28 THE SLIDE RULE:
CONTINUED MULTIPLICATION AND DIVISION.
By combining the rules for multiplication and division, we can
readily evaluate expressions of the form - x x - x " = x . The
simplest case, a * c can be solved by one setting of the slide.*
Take as an example, — - — = 102. Setting 8*5 on C to 14*45 on
o*0
D, we can, if desired, read 1*7 on D under 1 on C, as the quotient.
However, we are not concerned with this, but require its multipli-
cation by 60, and the slide being already set for this operation , we
at once read under 60 on C the result, 102, on D. The figures in
the answer are obvious.
When there are more factors to take into account, we place the
cursor over 102 on D, bring the next divisor on C to the cursor,
move the cursor to the next multiplier on C, bring the next divisor
on C to the cursor, and so on, until all the factors have been dealt
with. Note that only the first factor and the result are read on
D ; also that the cursor is moved for multiplying and the slide for
dividing.
Number of Digits in Result in Combined Multiplication and
Division. — For those who use rules the author's method of deter-
mining the decimal point in combined multiplication and division
may be used. Each time multiplication is performed with the slide
projecting to the right, make a - mark ; each time division is
effected with the slide to the right, make a I mark ; but allow
the 1 marks to cancel the - marks as far as they will. Subtract
the sum of the digits in the denominator from the sum of digits in
the numerator, and to this difference add any uncancelled memo-
marks, if of 1 character, or subtract them if of - character.
435 x 29*4 x 51 = 32_ = 1468>
27x3*83x10*5x1*31
Set 27 on C to 43*5 on D, and as with this division the
slide is to the right, make the first I mark. Bring cursor
to 29*4 on C, and' as in this multiplication the slide is to
the right, make the first - mark, cancelling as shown.
*The possible need for traversing the slide, to change the indices,
when using the C and D scales, is not considered as a setting.
A PRACTICAL MANUAL 29
Setting 3*83 on C to the cursor, requires the
second I mark, which, however, is cancelled in
turn by the multiplication by 51. The division
by 10*5 requires the third I mark, and after multi-
plying by 32 (requiring no mark) the final division
by 1*31 requires the fourth I mark. Then, as there
are 8 numerator digits, 6 denominator, and 2 un-
cancelled memo-marks (which, being I, are additive)
we have
Number of digits in result = 8-6 + 2 = 4.
Had the uncancelled marks been - in character, the number of
digits would have been 8-6-2 = 0.
For quantities less than O'l the digit place numbers will be
negative. The troublesome addition of these may be avoided by
transferring them to the opposite side and treating them as
2 4
™ 0-00356 x 27-1x0-08375 OQQ
positive. Thus: — ^ ., ^ — — - - — ——— = 288.
F 0-1426x9-85x0-00002
2 1 1
The first numerator, 000356, has -2 digits. Note this by
placing 2 beloiv the lower line as shown. 27'1 has 2 digits ; place 2
over it. 0*08375 has — 1 digit ; hence place 1 below the lower line.
The first denominator has no digits ; the second, 9*85, has 1 digit :
hence place 1 under it. 0*00002 has -4 digits ; place 4 above the
upper line. The sum of the top series is 2 + 4 = 6 ; of the bottom
series 2 + 1 + 1 = 4. Subtracting the bottom from the top, we have
6-4=2 digits, to which 1 has to be added for an uncancelled
memo-mark, and the result is read as 288.
Moving the decimal point often facilitates matters. Thus,
32*4x0-98x432x0-0217
4*71 x 0-175 x 0-00000621 x 41^000
is much more convenientlv dealt
•xi . a 32*4x9-8x432x2-17
with when re-arranged as — — — — — — =141.
8 4-71 x 17-5x6-21x4-12
To determine the number of figures in the result by rough
cancelling and mental calculation, we note that 4*71 enters 432
about 100 times ; 9*8 enters 17*5 about 2 ; 6'21 into 32'4 about 5 ;
and 2*17 into 4*12 about 2. This gives —= 125, showing that
the result contains 3 digits. From the slide rule we read 141,
which is therefore the result sought.
30 THE SLIDE RULE:
The occasional traversing of the slide through the rule, to
interchange the indices — a contingency which the use of the C and
D scales always involves— may often be avoided by a very simple
6*19 x 31'9 x 422
expedient. Such an example as — — — — ^-^=3*93 is some-
1120 x o'oo x 2*09
times cited as a particularly difficult case. Working through the
expression as given, two traversings of the slide are necessary ;
but by taking the factors in the slightly different order,
6*19x31-2x422 ., , ., . .- ,- - , .
— — — — — - — TT7Si$ so Mat the significant figures of each pair are more
8*86 x 2*09 x 1120 J r -
nearly alike, we not only avoid any traversing the slide, but we
also reduce the extent to which the slide is moved to effect the
several divisions.
o u axb axbx cxdxe n ,
Such cases as — or ? really resolve
cxdxexfxg fxg
, i i • , axbxlxlxl j axbxcxdx e -, , »
themselves into , — and , > but, of course,
cxdxexfxg fxgx\x\x\
if rules are used to locate the decimal point, the l's so (mentally) in-
troduced are not to be counted as additional figures in the factors.
MULTIPLICATION AND DIVISION WITH THE
SLIDE INVERTED.
If the slide be inverted in the rule but with the same face
uppermost, so that the O scale lies adjacent to the A scale, and the
right and left indices of the slide and rule are placed in coincidence,
we find the product of any number on D by the coincident number
on O (readily referred to each other by the cursor) is always 10.
Hence, by reading the numbers on O as decimals, we have over
any unit number on D, its reciprocal on O. Thus 2 on D is found
opposite 0'5 on ; 3 on D opposite to 0*333 ; while opposite 8 on
O is 0*125 on D, etc. The reason of this is that the sum of the
lengths of the slide and rule corresponding to the factors, is
always equal to the length corresponding to the product — in this
case, 10.
It will be seen that if we attempt to apply the ordinary rule
for multiplication, with the slide inverted, we shall actually be
multiplying the one factor taken on D by the reciprocal of the
other taken on O. But multiplying by the reciprocal of a
is equivalent to dividing by that number, and dividing a factor by
the reciprocal of a number is equivalent to multiplying by that
A PRACTICAL MANUAL 31
number. It follows that with the slide inverted the operations of
multiplication and division are reversed, as are also the rules for
the number of digits in the product and the position of the decimal
point. Hence, in multiplying with the slide inverted, we place
(by the aid of the cursor) one factor on opposite the other factor
on D, and read the result on D under either index of 0. It follows
that with the slide thus set, any pair of coinciding factors on
and D will give the same constant product found on D under the
index of 0. One useful application of this fact is found in select-
ing the scantlings of rectangular sections of given areas or in
deciding upon the dimensions of rectangular sheets, plates, cisterns,
etc. Thus by placing the index of to 72 on D, it is readily seen
that a plate having an area of 72 sq. ft. may have sides 8 by 9 ft.,
6 by 12, 5 by 14'4, 4 by 18, 3 by 24, 2 by 36, with innumerable
intermediate values. Many other useful applications of a similar
character will suggest themselves.
PEOPOETION.
With the slide in the ordinary position and with the indices of
the C and D scales in exact agreement, the ratio of the correspond-
ing divisions of these scales is 1. If the slide is moved so that
1 on C agrees with 2 on D, we know that under any number n on
C is n x 2 on D, so that if we read numerators on C and de-
nominators on D we have
_C 11-5234
Dl 2 3 4 6 8*
In other words, the numbers on D bear to the coinciding numbers
on C a ratio of 2 to 1. Obviously the same condition will obtain
no matter in what position the slide may be placed. The rule for
proportion, which is apparent from the foregoing, may be expressed
as follows : —
Rule for Proportion. — Set tJie first term of a proportion on the
C scale to the second term on the D scale, and opposite the third term
on the C scale read the fourth term on the D scale.
Ex. —Find the 4th term in the proportion of 20 : 27 : : 70 : x.
Set 20 on C to 27 on D, and opposite 70 on C read 94 "5 on D.
™ C 20 70
ThuS -D— 27 94*'
It will be evident that this is merely a case of combined
20 x 70
multiplication and division of the form, — — — =94*5. Hence,
32 THE SLIDE RULE:
given any three terms of a proportion, we set the 1st to the 2nd,
or the 3rd to the 4th, as the case may be, and opposite the other
given term read the term required.*
Thus, in reducing vulgar fractions to decimals, the decimal
3
equivalent of — is determined by placing 3 on C to 16 onD, when
16
over the index or 1 of D we read 0'1875 on C. In this case the
terms are 3 : 16 : : x : 1. For the inverse operation — to find a
vulgar fraction equivalent to a given decimal — the given decimal
fraction on C is set to the index of D, and then opposite any
denominator on D is the corresponding numerator of the fraction
onC.
If the index of C be placed to agree with 3*1416 on D, it will
be clear from what has been said that this ratio exists throughout
between the numbers of the two scales. Therefore, against any
diameter of a circle on C will be found the corresponding circum-
ference on D. In the same way, by setting 1 on C to the appropriate
conversion factor on D, we can convert a series of values in one
denomination to their equivalents in another denomination. In
this connection the following table of conversion factors will be
found of service. If the A and B scales are used instead of the
C and D scales, a complete set of conversions will be at once
obtained. In this case, however, the left-hand A and B scales
should be used for the initial setting, any values read on the right-
hand A or B scales being read as of tenfold value. With the C and
D scales a portion of the one scale will project beyond the other.
To read this portion of the scale, the cursor or runner is brought
to whichever index of the C scale falls within the rule, and the
slide moved until the other index of the C scale coincides with
the cursor, when the remainder of the equivalent values can then
be read off. It must be remembered that if the slide is moved in
the direction of notation (to the right), the values read thereon
have a tenfold greater value ; if the slide is moved to the left, the
readings thereon are decreased in a tenfold degree. Although
preferred by many, in the form given, the case is obviously one of
multiplication, and is so treated in the Data Slips at the end of
the book.
* The reader may be reminded that cross-multiplication of the
factors in any such slide rule setting will give a constant product, e.g. ,
20x94-5 = 27x70.
A PRACTICAL MANUAL
33
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THE SLIDE RULE:
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A PRACTICAL MANUAL
35
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36 THE SLIDE RULE:
Inverse Proportion.— If "more" requires "less," or "lew"
requires " more," the case is one of inverse proportion, and
although it will be seen that this form of proportion is quite
readily dealt with by the preceding method, the working is
simplified to some extent by inverting the slide so that the C
scale is adjacent to the A scale. By the aid of the cursor, the
values on the inverted C (or 0) scale, and on the D scale, can be
then read off. These will now constitute a series of inverse ratios.
For example, in the proportion
8 ~~4
D 1-5 3
the 4 on the scale is brought opposite 3 on D, when under 8 on
is found 1*5 on D.*
GENERAL HINTS ON THE ELEMENTARY USES
OF THE SLIDE RULE.
Before the more complex operations of involution, evolution, etc.,
are considered, a few general hints on the use of the slide rule
for elementary operations may be of service, especially as these
will serve to enforce some of the more important points brought
out in the preceding sections.
Always use the slide rule in as direct a light as possible.
Study the manner in which the scales are divided. Follow the
graduations of the C and D scales from 1 to 10, noting the values
given by each successive graduation and how these values change
as we follow along to the right. Do the same with the two halves
of the A and B scales and note the difference in the value of the
subdivisions, due to the shorter scale-lengths.
Practise reading values by setting 1 on C to some value on D
arid reading under 2, 3, 4, etc., on C, checking the readings by
mental arithmetic. To the same end, find squares, square roots,
etc., comparing the results with the actual values as given in tables.
Practise setting both slide and cursor to values taken at random.
Aim at accuracy ; speed will come with practice.
* In this case cross-dividing gives a constant quotient, e.g.,
8-r3 = 4-fl'5. Since the upper scale is now a scale of reciprocals, the
ratio is really §■ * * •
J D 1-5 3
A PRACTICAL MANUAL 37
When in doubt as to any method of working, verify by making
a simple calculation of the same form.
Follow the orthodox methods of working until entirely confident
in the use of the instrument, and even then do not readily make a
change. If any altered procedure is adopted, first work a simple
case and guard carefully against unconsciously lapsing into the
usual method during the operation.
Unless the calculation is of a straightforward character, time
taken in considering how best to attack . it (rearranging the
expression if desirable) is generally time well spent.
In setting two values together, set the cursor to one of them
on the rule, and bring the other, on the slide, to the cursor line.
In multiplying factors, as 57 x 0*1256, take the fractional value
first. It is easier to set 1 on C to 1256 on D and read under 57 on
C, than to reverse the procedure. When both values are eye-
estimated, set the cursor to the second factor on C and read the
result on D, under the cursor line.
In continuous operations avoid moving the slide further than
necessary, by taking the factors in that order which will keep the
scale readings as close together as possible.
SQUAEES AND SQUAKE EOOTS.
We have seen that the relation which the upper scales bear to
the lower set is such that over any number on D is its square on
A, and, conversely, under any number on A is its square root on
D, the same remarks applying to the C and B scales' on the slide.
Taking the values engraved on the rule, we have on D, numbers
lying between 1 and 10, and on A the corresponding squares ex-
tending from 1 to 100. Hence the squares of numbers between 1
and 10, or the roots of numbers between 1 and 100, can be read off
on the rule by the aid of the cursor. All other cases are brought
within these ranges of values by factorising with powers of 10, as
before explained.
The more practical rule is the following : —
To Find the Square of a Number, set the cursor to the number
on D and read the required square on A under the cursor. The
rule for
The Number of Digits in a Square is easily deducible from the rule
for multiplication. If the square is read on the left scale of A, it
38 THE SLIDE RULE:
will contain twice the number of digits in the original number
1 ; if it is read on the right scale of A, it will contain twice the
number of digits in the original number.
Ex. — Find the square of 114.
Placing the cursor to 114 on D, it is seen that the coinciding
number on A is 13. As the result is read off on the left scale of A,
the number of digits will be (3 x 2)-l = 5, and the answer is read
as 13,000. The true result is 12,996.
Ex.— Find the square of 0*0093.
The cursor being placed to 93 on D, the number on A is found
to be 865. The result is read on the right scale of A, so the
number of digits = —2x2= —4, and the answer is read as 0'0000865
[0-00008649].
Square Root. — The foregoing rules suggest the method of pro-
cedure in the inverse operation of extracting the square root of a
given number, which will be found on the D scale opposite the
number on the A scale. It is necessary to observe, however, that
if the number consists of an odd number of digits, it is to be taken
on the left-hand portion of the A scale, and the number of digits
in the root- , N being the number of digits in the original
A
number. When there is an even number of digits in the number,
it is to be taken on the right-hand portion of the A scale, and
the root contains one-half the number of digits in the original
number.
Ex. — Find the square root of 36,500.
As there *is an odd number of digits, placing the cursor to
365 on the l.h. A scale gives 191 on D. By the rule there are
- = = 3 digits in the required root, which is therefore
read as 191 [191 05].
Ex.— Find J 0*0098.
Placing the cursor to 98 on the right-hand scale of A (since -2
is an even number of digits), it is seen that the coinciding number
on D is 99. As the number of digits in the number is — 2, the
number of digits in the root will be— — = —1. It will therefore
be read as 0*099 [0*09899 + ].
Ex.— Find ,/ 0*098.
The number of digits is — 1, so under 98 on the left scale of A,
A PRACTICAL MANUAL 39
we find 313 on D. By the rule the number in the root will
be ~ 1 + 1 = 0, and the root is therefore read as 0313 [0313049 + ].
Ex.— Find ^ 0*149.
As the number of digits (0) is even, the cursor is set to 149 on
the right-hand scale of A, giving 386 on D. By the rule, the
number of digits in the root will be r — 0, and the root will be
read as 0-386 [0-38605+].
Another method of extracting the square root, by which more
accurate readings may generally be obtained, is by using the C and
D scales only, with the slide inverted. If there is an odd number
of digits in the number, the right index, or if an even number of
digits the left index, of the inverted scale O is placed so as to
coincide with the number on D of which the root is sought. Then
with the cursor, the number is found on D which coincides with
the same number on O, which number is the root sought.
Ex.— Find V22-2.
Placing the left index of O to 222 on D, the two equal coincid-
ing numbers on O and D are found to be 4*71.
Note that under the cursor line we have the original number,
22'2, on A, and from this the number of digits in the root is
determined as before.
The plan of finding the square of a number by ordinary multi-
plication is often very convenient: The inverse process of finding
a square root by trial division is not to be recommended.
To obtain a close value of a root or to verify one found in the
usual way, the author has, on occasion, adopted the following
plan : — Set 1 (or 10) on B to the number on the A scale (l.h. or
r.h. as the case may require), and bring the cursor to the number
on D. If the root found is correct, the readings on C under
the cursor and on D under the index of C, will be in exact agree-
ment.
If 1 on B is placed to a number n on the l.h. A scale, the student
will note that while root n is read on D under 1 on C, the root of
10 n is read on D under 10 on B. Hence, if preferred, the number
can be taken always on the first scale of A and the root read under
1 or 10 on B, according to whether there is an odd or even number
of digits in the number. Obviously the second root is the first
multiplied by „J 10.
40 TEE SLIDE RULE I
CUBES AND CUBE BOOTS.
In raising a number to the third power, a combination of the
preceding method and ordinary multiplication is employed.
To Find the Cube of a Number. — Set the l.h. or r.h. index of
C to the number on D, and opposite the number on the left-hand
scale of B read the cube on the l.h. or r.h. scale of A.
By this rule four scales are brought into requisition. Of these,
the D scale and the l.h. B scale are always employed, and are to
be read as of equal denomination. The values assigned to the l.h.
and r.h. scales of A will be apparent from the following considera-
tions.
Commencing with the indices of C and D coinciding, and
moving the slide to the right, it will be seen that, working in
accordance with the above rule, the cubes of numbers from 1 to
2*154 (= VlO) will be found on the first or l.h. scale of A. Moving
the slide still farther to the right, we obtain on the r.h. A scale
cubes of numbers from 2*154 to 4*641 (or V~10~to %J1Q0). Had we
a third repetition of the l.h. A scale, the l.h. index of C could be
still further traversed to the right, and the cubes of numbers from
4*641 to 10 read off on this prolongation of A. But the same end
can be attained by making use of the r.h. index of C, when,
traversing the slide to the right as before, the cubes of numbers
from 4*641 to 10 on D can be read off on the l.h. A scale over the
corresponding numbers on the l.h. B scale. Hence, using the
l.h. index of C, the readings on the l.h. A scale may be regarded
comparatively as units, those on the r.h. A scale as tens ; while
for the hundreds we again make use of the l.h. A scale in con-
junction with the right-hand index of C.
By keeping these points in view, the number of digits in the
cube (N) of a given number (n) are readily deduced. Thus, if the
units scale is used, N = 3 n - 2 ; if the tens scale, N = 3 n - 1 ; while if
the hundreds scale be used, N = 3 n. Placed in the form of rules : —
N = 3 n— 2 when the product is read on the l.h. scale of A with
the slide to the right (units scale).
N = 3 n-\ when the product is read on the r.h. scale of A ;
slide to the right (tens scale).
N = 3 n when the product is read on the l.h. scale of A with the
slide to the left (hundreds scale).
A PRACTICAL MANUAL 41
With decimals the same rule applies, but, as before, the number
of digits must be read as - 1, — 2, etc., when one, two, etc., cyphers
follow immediately after the decimal point.
Ex.— Find the value of 1*4 3 .
Placing the l.h. index of C to 1'4 on D, the reading on A
opposite 1*4 on the l.h. scale of B is found to be about 2745
[2-744].
Ex.— Find the value of 26*4 3 .
Placing the l.h. index of C to 26 '4 on D, the reading on A
opposite 26*4 on the l.h. scale of B is found to be about 18,400
[18,399744].
Ex.— Find the value of 73 3 .
In this case it becomes necessary to use the r.h. index of C,
which is set to 7*3 on D, when opposite 7 '3 on the l.h. scale of B
is read 389 [389'017] on A.
Ex.— Find the value of 0'073 3 .
From the setting as before it is seen that the number of digits
in the number must be multiplied by 3 . Hence, as there is — 1 digit
in 0*073, there will be —3 in the cube, which is therefore read
0-000389.
The last two examples serve to illustrate the principle of
factorising with powers of 10. Thus
0*073 = 7-3 x 10~ 2 ; 0'073 3 =7-3 3 x (10" 2 ) 3 =389 x 10' 6 =0-000389.
Cube Root (Direct Method). — One method of extracting the cube
root of a number is by an inversion of the foregoing operation.
Using the same scales, the slide is moved either to the right or left
until under the given number on A is found a number on the L.n.
B scale, identical with the member simultaneously found on D under
the right or left index of O. This number is the required cube
root.
From what has already been said regarding the combined use
of these scales in cubing, it will be evident that in extracting the
cube root of a number, it is necessary, in order to decide which
scales are to be used, to know the number of figures to be dealt
with. We therefore (as in the arithmetical method of extraction)
point off the given number into sections of three figures each,
commencing at the decimal point, and proceeding to the left for
numbers greater than unity, and to the right for numbers less
than unity. Then if the first section of figures reading from the
left consists of —
42 THE SLIDE RULE:
1 figure, the number will evidently require to be taken on
what we have called the " units" scale— i.e., on the l.h. scale of A,
using the l.h. index of C.
If of 2 figures, the number will be taken on the " tens M
scale — i.e., on the r.h. scale of A, using the l.h. index of C.
If of 3 figures, the number will be taken on the "hundreds"
scale — i.e., on the l.h. scale of A, using the r.h. index of C.
To determine the number of digits in cube roots it is only
necessary to note that when the number is pointed off into sections
as directed, there will be one figure in the root for every section
into which the number is so divided, whether the first section con-
sists of 1, 2, or 3 digits.
Of numbers wholly decimal, the cube roots will be decimal, and
for every group of three O's immediately following the decimal
point, one will follow the decimal point in the root. If neces-
sary, O's must be added so as to make up even multiples of 3
figures before proceeding to extract the root. Thus 0*8 is to
be regarded as 0*800, and 0*00008 as 0*000080 in extracting cube
roots.
Ex.— Find Vl4,000.
Pointing the number off in the manner described, it is seen
that there are two figures in the first section — viz., 14. Setting the
cursor to 14 on the r.h. scale of A, the slide is moved to the right
until it is seen that 241 on the l.h. scale of B falls under the
cursor, when 241 on D is under the l.h. index of C. Pointing
14,000 off into sections of three figures, we have 14 000 — that is,
two sections. Therefore, there are two digits in the root, which
in consequence will be read 24*1 [24*1014 + ].
Ex.— Find IJ&V52.
As the divisional section consists of three figures, .we use the
"hundreds" scale. Setting the cursor to 0*162 on the l.h. A
scale, and using the r.h. index of C, we move the slide to the left
until under the cursor 0*545 is found on the l.h. B scale, while the
r.h. index of C points to 0*545 on D, which is therefore the cube
root of 0*102.
Ex.— Find y'00002.
To make even multiples of 3 figures requires the addition of
00 ; we have then 200, the cube root of which is found to be about
5*85. Then, since the first divisional group consists of O's, one
will follow the decimal point, giving V0 T W02=0*0585 [0*05848].
A PRACTICAL MANUAL 43
Cube Root (Inverted Slide Method). — Another method of extract-
ing the cube root involves the use of the inverted slide. Several
methods are used, but the following is to be preferred : — Set the
l.h. or r.h. index of the slide to the number on A, and the number
on U (i.e., R inverted), which coincides with the same number on JJ,
is the required root.
Setting the slide as directed, and using first the l.h. index of
the slide and then the r.h. index, it is always possible to find three
pairs of coincident values. To determine which of the three is the
required result is best shown by an example.
Ex.— Find 1/^%/bO, and %/W).
Setting the r.h. index of the slide to 5 on A, it is seen that
1*71 on D coincides with 1*71 on 9. Then setting the l.h. index
to 5 on A, further coincidences are found at 3*68 and at 7*93,
the three values thus found being the required roots. Note that
the first root was found on that portion of the D scale lying under
1 to 5 on A ; the second root on that portion lying under 5 to 50
on A ; and the third root on that portion of D lying under 50 to
100 on A. In this connection, therefore, scale A may always be
considered to be divided into three sections — viz., 1 to n, n to 10 n,
and 10 n to 100. For all numbers consisting of 1, 1+3, 1 +6, 1 + 9
— i.e., of 1, 4, 7, 10, or — 2, —5, etc., figures— the coincidence under
the first section is the one required. If the number has 2, 5, 8, or
— 1, —4, —7, etc., figures, the coincidence under the second section
is correct, while if the number has 3, 6, 9, or 0, — 3, etc., figures,
the coincidence under the last section is that required. The
number of digits in the root is determined by marking off the
number into sections, as already explained.
Cube Root (PickwortKs Method). — One of the principal objections
to the two methods described is the difficulty of recollecting which
scales are to be employed and with which index of the slide they
are to be used. With the direct method another objection is that
the readings to be compared are often some distance apart, the
maximum distance intervening being two-thirds of the length of
the rule. To carry the eye from one to another is troublesome
and time-taking. With the inverted scale method the reading of
a scale reversed in direction and with the figures inverted is also
objectionable.
With the author's method these objections are entirely obviated.
The same scales and index are always used, and are read in their
E
44 THE SLIDE RULE:
normal position. The three roots of n, 10 n and 100 n are given
with one setting and appear in their natural sequence, no travers-
ing of the slide being needed. The readings to be compared are
always close together, the maximum distance between them being
one-sixth of the length of the rule. The setting is always made in
the earlier part of the scales where closer readings can be obtained,
and finally, if desired, the result may be readily verified on the
lower scales by successive multiplication.
For this method two gauge points are required on C. To con-
veniently locate these, set 53 on C to 246 on D ; join 1 on D to
1 on A with a straight-edge and with a needle point draw a short
fine line on C. Set 246 on C to 53 on D, and repeat the process
at the other end of the rule. The gauge points thus obtained
will be at 2*154 and 4"641, and should be marked %/T0 and \jTO0
respectively.*
Ex.— Find %/¥86, V^B and 5/286.
Set cursor to 2'86 on A and drawing the slide to the right
find 1*42 under 1 on C, when 1'42 on B is under the cursor. Then
reading under 1, VlO and VlOO, we have
V2 : 86 = l-42; %/28r6 = Z'06 and 5/286 = 6*59.
It will be seen that factorising with powers of 10, we multiply
the initial root by 2/10 and V100. Obviously the three roots will
always be found on D, in their natural order and at intervals of
one-third the length of the rule. The number of digits in the
roots of numbers which do not lie between 1 and 1000, is found as
before explained.
In any method of extracting cube roots in which the slide has
to be adjusted to give equal readings on B and D, the author lias
found it of advantage to adopt the following plan : — The cursor
being set to, say, 4*8 on A, bring a near main division line on B,
as 1*7, to the cursor ; then 1 on C is at 1*68 on D. The difference
in the readings is two small divisions on D, and moving the slide
forward by one-third the space representing this difference, we obtain
1*687 as the root required. With a little practice it is possible to
obtain more accurate results by this method than by comparing
the reading on D with that on the less finely-graded B scale.
* These lines should not be brought to the working edge of the scale but
should terminate in the horizontal line which forms the border of the
finer graduations, their value being read into the calculation by nu-ans ol
the cursor (see page 55).
A PRACTICAL MANUAL 45
MISCELLANEOUS POWERS AND ROOTS.
In addition to squares and cubes, certain other powers and
roots may be readily obtained with the slide rule.
Two-thirds Power. — The value of N» is found on A over %JN on
D. The number of , digits is decided by the rule for squares,
working from the number of digits in the cube root. It will often
be found preferable to treat N* as N-f %fN, as in this way the
magnitude of the result is much more readily appreciated.
Three-two Power. — N' 5 " can be obtained by cubing the square
root, deciding the number of digits in each process. For the
reason just given, it is preferable to regard N 5 as N x ^/N.
Fourth Power. — For N 4 set the index of C to N on D and over
N on C read N 4 on A ; or find the square of the square of N,
deciding the number of digits at each step.
Fourth Boot. — Similarly for VN, take the square root of the
square root.
Four-third Power. — N^=N 1,33 (useful in gas-engine diagram
calculations) is best treated as N x %/N.
Other powers can be found by repeated multiplication. Thus
setting 1 on B to N on A, we have on A, N 2 over N ; N 3 over N 2 ;
N 4 over N 3 ; N 5 over N 4 , etc. In the same way, setting N on B
to N on D, we can read such values as N*, N», etc.
POWERS AND ROOTS BY LOGARITHMS.
For powers or roots other than those of the simple forms
already discussed, it is necessary to employ the usual logarithmic
process. Thus to find a n =x, we multiply the logarithm of a by n,
and find the number x corresponding to the logarithm so obtained.
Similarly, to find 1 tja=x we divide the logarithm of a by n, and
find the number x corresponding to the resulting logarithm.
The Scale of Logarithms. — Upon the back of the slide of the
Gravet and similar slide rules there will be found three scales.
One of these— usually the centre one — is divided equally through-
out its entire length, and figured from right to left. It is some-
times marked L, indicating that it is a scale giving logarithms.
The whole scale is divided primarily into ten equal parts, and each
of these subdivided into 50 equal parts. In the recess or notch in
the right-hand end of the rule is a reference mark, to which any
of the divisions of this evenly-divided scale can be set.
46 THE SLIDE RULE:
As this decimally-divided scale is equal in length to the
logarithmic scale T>, and is figured in the reverse direction, it
results that when the slide is drawn to the right so that the l.h.
index of C coincides with any number on D, the reading on the
equally-divided scale will give the decimal part of the logarithm
of the number taken on D. Thus if the l.h. index of C is placed
to agree with 2 on D, the reading of the back scale, taken at the
reference mark, will be found to be 0*301, the logarithm of 2. It
must be distinctly borne in mind that the number so obtained is
the decimal part or mantissa of the logarithm of the number, and
that to this the characteristic must be prefixed in accordance with
the usual rule — viz., The integral part, or characteristic of a logarithm
is equal to the number of digits in the number, minus 1. If the
member is wholly decimal, the characteristic is equal to the number of
cyphers following the decimal point, plus 1. In the latter case the
characteristic is negative, and is so indicated by having the minus
sign written over it.
To obtain any given power or root of a number, the operation
is as follows : — Set the l.h. index of C to the given number on
D, and turning the rule over, read opposite the mark in the notch
at the right-hand end of the rule, the decimal part of the logarithm
of the number. Add the characteristic according to the above
rule, and multiply by the exponent of the power, or divide by
the exponent of the root. Place the decimal fart of the resultant
reading, taken on the scale of equal parts, opposite the mark in
the aperture of the rule, and read the answer on D under the l.h.
index of C, pointing off the number of digits in the answer in
accordance with the number of the characteristic of the resultant.
Ex.— Evaluate 36 1 ' 414 .
Set 1 on C to 36 on D and read the decimal part of log. 36 on
the scale of logarithms on the back of the slide. This value is
found to be 0*556. As there are two digits in the number, the
characteristic will be 1 ; hence log. 36 = 1*556. Multiply by 1111,
using the C and D scales, and obtain 2*2 as the log. of the result.
Set the decimal part, 0*2, on the log. scale to the mark in the notch
at the end of the rule ami read 1585 on D under 1 on C. Since
the log. of the result has a characteristic 2, there will be 3 dibits
in the result, which is therefore read as 158*5,
This example will suffice to show the method of obtaining the
wth power or the n\\\ root of any number.
A PRACTICAL MANUAL
47
OTHER METHODS OF OBTAINING POWERS
AND ROOTS.
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>G7 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", 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.
The converse procedure for
obtaining the ?ith root of a
number N will obviously re-
solve itself into obtaining -th
° n
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 y the author has found the following device of assistance : — On a
piece of thin transparent celluloid a line OC is drawn (Fig. 11) and
OC .
'or,
convenient to make OE=1-10 on the A scale, so that assuming
we require a series of values of i; 1 * 35 , O B would be 12*5 cm. and O C,
16*875 cm. On these lines semi-circles are drawn as shown, both
passing through the point 0.
Fig. 11.
48 THE SLIDE HULK:
Applying this cursor to the upper scales so that the point O is
on 1 and the semi-circle OMB passes through v on A, the larger
semi-circle will give on A the value of v n . Thus for pv 1l = 39'b x4U 1- •',
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 OMB p;i
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
Fig. 12.
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.
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
A PRACTICAL MANUAL
49
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
AON 2 and N*N 2 the length of AO is twice that of Nf, it results
that A N- = 2AN. In general, then, to find the nth 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" on A. The inverse proceeding for finding the nth root will be
obvious.
Fig. 13.
An advantage offered by this and analogous methods of obtain-
ing 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.
COMBINED OPEBATIONS.
Thus far the various operations have been separately considered,
and we now pass on to a consideration of the methods of working for
solving the various formulae met with in technical calculations. We
propose to explain the methods of dealing with a few of the more
generally used expressions, as this will suffice to suggest the proced-
ure in d ealing with other and more intricate calculations. In solving
50 THE SLIDE MILE!
the following problems, both the upper and lower scales arc d
and the relative value of the several scales must be observed
/745
throughout. Thus, in solving such an expression as / — — = 6*86,
the division is first effected by setting 15*8 on B to 745 on A.
From the relation of the two parts of the upper scales (page 37)
we know that such values as 7 '45, 745, etc., will be taken on the
left-hand A and B scales, while values as 15'8, 1580, etc., will be
taken on the right-hand A and B scales. Hence, 15'8 on the r.h.
B scale is set to 745 on the l.h. A scale, and the result read on D
under the index of C. Had both values been taken on the l.h. A and
B scales, or both on the r.h. A and B scales, the results would have
corresponded to^=^^ = 2'17,orto^=^/jl2 = 2-17, i.e, to
-= Hence if a wrong choice of scales has been made, we can
V 10
correct the result by multiplying or dividing by \/10 as the case
may require. If the result is read on D, set to it the centre
index (10) of B and read the corrected result under the index of C.
To solve axb 2 —x. Set the index of C to b on D, and over a
on B read x on A.
w
To solve ~y = x. Set b on B to a on D by using the cursor,
and over index of B read x on A.
To solve ~o = x. Set a on C to b on A, and over 1 on B read
x on A.
To solve aX - x - Set c on B to b on D, and over a on B
read x on A.
To solve (a x b) 2 = x. Set 1 on C to a on D, and over b on C
read x on A.
To solve (y ) =x. Set b on C to a on D, and over 1 on C
read x on A.
To solve <sjaxb==x. Set 1 on B to a on A, and under b on B
read x on D. _
To solve A j-x. Set 6 on B on a on A, and under 1 on C
read x on D,
A PRACTICAL MANUAL 51
To solve a -,=x. Set b on C to c on D and over a on B read x
c
on A.
j=x. Set & on B to a on A, and under c on C
To solve~r~=^. Set b on C to a on A, and under 1 on C
read x on D
To solve'
read x on D.
To solve ~t=x. Set 6 on B to a on D, and under 1 on C
read x on D.
To solve b sja=x. Set 1 on C to b on D, and under a on B
read x on D.
To solve \Ja?=x. Treat as a\/a.
To solve a\Jb i =x. Treat as a\/b x 5.
To solve ^L=x. Treat as *£*•
6 b
To solve /?=*. Treat as &? = /?x*.
V J>_ V& V *>
To solve A / =x. Set c on B to a on A, and under b on
B read x on D.
a x 6
To solve — xs =#. Set c on B to b on D, and under a on C
V c
read x on D.
To solve / — x. Set c on B to a on D, and under 6 on
B read x on D.
To solve =#. Set c on B to a on D, and over b on C
c
read x on A.
To solve ^Ll=x. Set c on C to b on A, and under a on C
read x on D.
To solve ( x ^ } J ~x. Set c on C to a on D, and over b on
B read x on A.
52 THE SLIDE RULE:
HINTS ON EVALUATING EXPRESSIONS.
As a general rule, the use of cubes and higher powers should be
avoided whenever possible. Thus, in the foregoing section, we
recommend treating an expression of the form ajb 3 as ax b x ,Jb\
the magnitudes of the values thus met with are more easily appre-
ciated by the beginner, and mistakes in estimating the large num-
bers involved in cubing are avoided.
Ex.— 7'3 x ^57^3140.
Set 1 on C to 57 on D ; bring cursor to 57 on B (r.ii., since 57
has an even number of digits) ; bring 1 on C to cursor, and under
7 '3 on C read 3140 on D. As a rough estimate we have \/57,
about 8 ; 8 x 57, about 400 ; 400 x 7, gives 2800, showing the result
consists of 4 figures.
An expression of the form a %/b\ or ab%, is better dealt with by
rearranging as a x — _.
_%/b
Ex.— 3'64 V4*32 2 = 9*65.
Set cursor to 4*32 on A, and move the slide until 1*63 is found
simultaneously under the cursor on B and on D under 1 on C ;
bring cursor to 1 on C ; 4 '32 on C to cursor, and over 3*64 on D
read 9*65 on C. (Note that in this case it is convenient to read the
answer on the slide ; see page 22). From the slide rule we know
3^/4-32 = about 1*6 ; this into 4*32 is roughly 3 ; 3*64 x 3 is about 10,
showing the answer to be 9*65.
Similarly products of the form ax^ are best dealt with as
a x b x %/b.
Factorising expressions sometimes simplifies matters, as, for in-
stance, in x*— y*=(x' 1 +y-){x 2 — y~). Here, working with the fourth
powers involves large numbers and the troublesome determination
of the number of digits in each factor ; but squares are read on
the rule at once, the number of digits is obvious, and, in general,
the method should give a more accurate result. Take the ex-
pression, D 1= / — giving the diameter T> 1 of a solid shaft
equal in torsional strength to a hollow shaft whose external and
internal diameters are D and d respectively. Rearranging as
*-V
^ '- and taking, as an example, D=15 in.
A PRACTICAL MANUAL 53
and d=7 in., we have D 2 + d 2 =274 and T> 2 -dr=176; hence
D 1= 7 ^ — =V3210= 1475 in.
Reversed Scale Notation. — With expressions of the form 1 — ,r,
or 100— x, it is often convenient to regard the scales as having
their notation reversed, i.e., to read the scale backwards. When
this is done the D scale is read as shown on the lower line —
Direct Notation 123456789 10
D Scale
Eeversed Notation 987654321
The new reading can be found by subtracting the ordinary reading
from 1, 10, 100, etc., according to the value assigned to the r.h.
index, but actually it is unnecessary to make this calculation, as
with a little practice it is quite an easy matter to read both the
main and subdivisions in the reversed order. Applications are
found in plotting curves, trigonometrical formulae, etc.
Ex. — Find the per cent, of slip of a screw propeller from
100-8=1^1
P XX
taking the speed, V, as 15 knots, the pitch of the propeller, P, as
27 ft. 6 in., and the revolutions per minute, R, as 60.
Set 27-5 on B to 10133 on A (N.B.— Take the setting near the
centre index of A) ; bring the cursor to 15 on B and 60 on B to
cursor. Reading the l.h. A scale backwards, the slip, S, =8 per
cent, is found on A over 10 on B.
Percentage Calculations. — To increase a quantity by x per cent,
we multiply by 100 +x ; to diminish a quantity by x per cent, we
multiply by 100— .r. Hence, to add x per cent., set 100 +x on C to
1 on D and read new values on D under original values on C. To
deduct x per cent, read the D scale backwards from 10 and set r.h.
index of C to x per cent, so read. Then read as before.
GAUGE POINTS.
Special graduations, marking the position of constant factors which
frequently enter into engineering calculations, are found on most
slide rules. Usually the values of 7r=3*1416 and - = 0'7854— the
"gauge points" for calculating the circumference and area of a
circle — are marked on the upper scales. The first should be given
54 THE SLIDE RULE!
on the lower scales also. Marks c and c 1 are sometimes found on
-and at 3'568= / — • These are
useful in calculating the contents of cylinders and are thus
derived : — Cubic contents of cylinder of diameter d and length 1 =
jd 2 l; substituting for 7 its reciprocal -, the formula becomes
d 2
- x 7, and by taking the square root of the fractional part we
/ d V
have ( . t19R J x I. This is now in a very convenient form, since
by setting the gauge point c on C to d on D, we can read over I on
B the cubic contents on A. This example indicates the principle
to be followed in arranging gauge points. Successive multiplica-
tion is avoided by substituting the reciprocal of the constant, thus
bringing the expression into the form - — , which, as we know,
can be resolved by one setting of the slide. The advantage of
dividing d before squaring is also evident. The mark c l -cx\/10
is used if it is necessary to draw the slide more than one-half its
length to the right.
A gauge point, M, at 31 83 = is found on the upper scales of
some rules. Setting this point on B to the diameter of a cylinder
on A, the circumference is read over 1 or 100 on B or the area of
the curved surface over the length on B.
As another example of establishing a gauge point, we will take
the formula for the theoretical delivery of pumps. If d is the
diameter of the plunger in inches, I the length of stroke in feet,
and Q the delivery in gallons, w T e have
7T 12
Q = d-x .xlx -—_. (N.B.— 277 cubic inches = 1 gallon.)
Multiplying out the constant quantities and taking its reciprocal,
we readily transform the statement into Q = or ( - - ) xl.
J ^ 29-4 \5'42/
Hence set gauge point 5'42 on C to d on I) and over length of
stroke in feet on B, read delivery in gallons per stroke on A ; <>r
over piston speed in feet per minute on B, read theoretical deliver/
in gallons per minute on A.
Several examples of gauge points will be found in the section
A PRACTICAL MANUAL 55
on calculating the weights of metal (see pages 59 and 60). In
most cases their derivation will be evident from what has been
said above. In the case of the weight of spheres, we have Vol.=
0*5236 <i 3 , and this multiplied by the weight of 1 cubic inch of the
material will give the weight W in lb. Hence for cast-iron, W =
0*5236 x (p x 0*26, which is conveniently transformed into W= _-^—-
as in the example on page 60.
With these examples no difficulty should be experienced in
establishing gauge points for any calculation in which constant
factors recur.
Marking Gauge Points. — The practice of marking gauge points
by lines extending to the working edge of the scale is not to be
recommended, as it confuses the ordinary reading of the scales.
Generally speaking, gauge points are only required occasionally,
and if they are placed clear of the scale to which they pertain,
but near enough to show the connection, they can be brought
readily into a calculation by means of the cursor. Usually there
is sufficient margin above the A scale and below the D scale for
various gauge points to be marked. Another plan consists in
cutting two nicks in the upper and lower edges of the cursor near
the centre and about Jin. apart. These centre pieces, when bent
out, form a tongue, which are in line with the cursor line and run
nearly in contact with the square and bevelled edges of the rule re-
spectively. A fine line in the tongue can then be set to gauge points
marked on these two edge strips, the ordinary measuring gradua-
tions being removed, if desired, by a piece of fine sand-paper.
For gauge points marked on the face of the rule, the author
prefers two fine lines drawn at 45° — thus, X — and crossing in the
exact point which it is required to indicate. With the " cross "
gauge point the meeting lines facilitate the placing of the cursor, and
an exact setting is readily made.* All lines should be drawn in
Indian ink with a very sharp drawing pen. For a more permanent
marking the Indian ink may be rubbed up in glacial acetic acid or
the special ink for celluloid may be used. If any difficulty is found
in writing the distinguishing signs against the gauge point, the
inscription may be formed by a succession of small dots made with
a sharp pricker.
* The same principle may be applied to the cursor.
56 THE SLIDE RULE:
EXAMPLES IN TECHNICAL CALCULATIONS.
In order to illustrate the practical value of the slide rule,
we now give a number of examples which will doubtless be
sufficient to suggest the methods of working with other formulae.
A few of the rules give results which are approximate only,
but in all cases the degree of accuracy obtained is well
within the possible reading of the scales. In many cases the
rules given may be modified, if desired, by varying the constants.
In most of the examples the particular formula employed will be
evident from the solution, but in a few of the more complicated
cases a separate statement has been given.
Mensuration, Etc.
Given the chord c of a circular arc, and the vertical height h,
to find the diameter d of the circle.
Set the height h on B to half the chord on D, and over 1 on B
read x on A. Then % + h—d.
Ex.— c = 6 ; h = 2 ; find d.
Set 2 on B to 3 on D, and over 1 on B read 4*5 on A. Then 4*5 + 2
= 6-5 = d.
Given the radius of a circle r, and the number of degrees n in
an arc, to find the length I of the arc.
Set r on C to 57 '3 on D, and over any number of degrees n on
D read the (approximate) length of the arc on C.
Ex-— r = 24 ; n = 30 ; find I.
Set 24 on C to 57 "3 on D, and over 30 on D read 12«56 = Z on C.
Given the diameter d of a circle in inches, to find the circum-
ference c in feet.
Set 191 on C to 50 on D, and under any diameter in inches on
C read circumference c, in feet on D.
Ex. — Find the circumference in feet of a pulley 17in. in diameter.
Set 191 on C to 50 on D, and under 17 on C read 4 -45ft. on D.
Given the diameter of a circle, to find its area.
Set 0'7854 onBto 10 (centre index) on A and over any diameter
on D read area on B.
When the rule has a special graduation line = 0*7854, on the
right-hand scale of B, set this line to the R.H, index of A and read
oil" as above. If only ir is marked, set this special graduation on
B to 4 on A.
A PRACTICAL MANUAL 57
On the C and D scales of some rules is a gauge point
marked c will be found indicating x / ~ = 1*128. In this case,
therefore, set 1 on C to gauge point c on D, and read area on A
as above. If the gauge point c' is used, divide the result by 10
Or set c on C, to diameter on D, and over index of B read area on A.
Cursors are supplied, having two lines ruled on the glass, the
4
interval between them being equal to — = 1*273 on the A scale. In
this case, if the right hand of the two cursor lines be set to the
diameter on D, the area will be read on A under the left-hand
cursor line. For diameters less than 1*11 it is necessary to set the
middle index of B to the l.h. index of A, reading the areas on the
l.h. B scale. The confusion which in general work is sometimes
caused by the use of two cursor lines might be obviated by making
the left-hand line in two short lengths, each only just covering the
scales.
Given diameter of circle d in inches, to find area a in square
feet.
Set 6 on B to 11 on A, and over diameter in inches on D read
area in square feet on B.
To find the surface in square feet of boiler flues, condenser
tubes, heating pipes, etc., having given the diameter in inches and
length in feet.
Find the circumference in feet as above and multiply by the
length in feet.
Ex. — Find the heating surface afforded by 160 locomotive boiler tubes
If in. in diameter and 12ft. long.
Set 191 on C to 50 on D ; bring cursor 1*75 on C, L.H. index of C to
cursor ; cursor to 12 on C ; 1 on C to cursor ; and under 160 on C read
880 sq. ft. of heating surface on D.
If the dimensions are in the same denomination and the rule has
a gauge point M at 31*83 f = ), set this mark on B to diameter of
cylinder on A, and read cylindrical surface on A over length on B.
To find the side s of a square, equal in area to a given rectangle
of length I and breadth b.
Set r.h. or l.h. index of B to I on A, and under b on B read 5
onD.
58 THE SLIDE RULE I
Ex. — Find the side of a square equal in area to a rectangle in which
J = 31ft. and 6 = 5ft.
Set the (r.h.) index of B to 31 on A, and under 5 on B read
1245ft. on D.
To find various lengths I and breadths b of a rectangle, to give
a constant area a.
Invert the slide and set the index of to the given area on D.
Then opposite any length I on find the corresponding breadth
b on D.
Ex. — Find the corresponding breadths of rectangular sheets, 16, 18,
24, 36, and 60ft. long, to give a constant area of 72 sq. ft.
Set the r.h. index of to 72 on D, and opposite 16. 18, 24, 36,
and 60 on read 4*5, 4, 3, 2, and l*2ft., the corresponding breadths
onD.
To find the contents in cubic feet of a cylinder of diameter d in
inches and length I in feet.
Find area in feet as before, and multiply by the length.
If dimensions are all in inches or feet, set the mark c ( = 1*128)
on C to diameter on D and over length on B, read cubic contents
on A.
To find the area of an ellipse.
Set 205 on C to 161 on D ; bring cursor to length of major
axis on C, 1 on C to cursor, and under length of minor axis on C
read area on D.
Ex. — Find the area of an ellipse the major and minor axes of which
are 16in. and 12in. in length respectively.
Set 205 on C to 161 on D ; bring cursor to 16 on C, 1 on C to
cursor, and under 12 on C read 150 '8in. on D.
To find the surface of spheres.
Set 3*1416 on B to r.h. or l.h. index of A, and over diameter
on D read by the aid of the cursor, the convex surface on B.
To find the cubic contents of spheres.
Set 1*91 on B to diameter on A, and over diameter on C read
cubic contents on A.
Weights of Metals.
To find the weight in lb. per lineal foot of square bars of metal.
Set index of B to weight of 12 cubic inches of the metal (/>.,
one lineal foot, 1 square inch in section) on A, and over the side of
the square in inches on read weight in lb. on A.
A PRACTICAL MANUAL
59
Ex. — Find the weight per foot length of 4£in. square wrought-iron
bars.
Set middle index of B to 3*33 on A, and over 4 J on C read 67*5
lb. on A.
(N.B. — For other metals use the corresponding constant in
column (2), below).
To find the weight in lb. per lineal foot of round bars.
Set r.h. or l.h. index of B to weight of 12 cylindrical inches of
the metal on A (column (4), below), and opposite the diameter of
the bar in inches on C, read weight in lb. per lineal foot on A.
Ex. — Find the weight of 1 lineal foot of 2in. round cast steel.
Set l.h. index of B to 2*68 on A, and over 2 on C read 10 '71b. on A.
To find the weight of flat bars in lb. per lineal foot.
Set the breadth in inches on C to — — — - ; — — of the
weight of 12 cub. in.
metal (column (3), below) on D, and above the thickness on D
read weight in lb. per lineal foot on C.
Ex. — Find the weight per lineal foot of bar steel, 4^in. wide and
fin. thick.
Set 4'5 on C to 0'294 on D, and over 0'625 on D read 9'561b. per
lineal foot on C.
To find the weight per square foot of sheet metal, set the
weight per cubic foot of the metal (col. 1) on C to 12 on D, and
Metals
Weight in
lb. per
cubic ft.
(2)
Weight of
12 cubic in.
(3)
1
(4)
Weight of 12
Wt. of 12 cub. in.
cylindrical
in.
Wrought iron . . .
Cast iron
Cast steel
Copper
480
450
490
550
168
520
710
462
430
450
3-33
3-125
3-40
3-82
1-166
3-61
4-93
321
2-98
3-125
300
0-320
0-294
0-262
0-085
0-277
0-203
0-312
0-335
0-320
2-62
2-45
2-68
3-00
Aluminium
Brass
0-915
2-83
Lead
3-87
Tin
2-52
Zinc (cast)
„ (sheet) . . .
2-34
2-45
above the thickness of the plate in inches on D read weight in lb.
per square foot on C.
p
60
THE SLIDE llULEl
Ex. — Find the weight in lb. per square foot of aluminium sheet fin.
thick.
Set 168 on to 12 on D, and over 375 on D read 5*251b. on C.
To find the weight of pipes in lb. per lineal foot.
Set mean diameter of the pipe in inches (i.e., internal diameter
plus the thickness, or external diameter minus the thickness) on C
to the constant given below on D, and over the thickness on D
read weight in lb. per lineal foot on C.
Metals.
Constant for
Pipes.
Constant for
Spheres.
Wrought iron
Oast iron
oooooo
o o OO H* o
O 00 CO CO O CD
•Pa OJ CO Ol to Cn
O -P* CO O O Cn
6 87
7-35
6-73
6-35
6-00
4-65
Steel
Brass
Copper
Lead
Ex. — Find the weight per foot of cast-iron piping 4in. internal
diameter and ^in. thick.
Set 4-5 on C to 0*102 on D, and over 0*5 on D read 22 'lib. on C,
the required weight.
To find the weight in lb. of spheres or balls, given the diameter
in inches. (W= 0*5236 d 3 x wt. of 1 cub. in. of material).
Set the constant for spheres (given above) on B to diameter in
inches on A, and over diameter on C read weight in lb. on A.
Ex. — Find the weight of a cast-iron ball 7£in. in diameter.
Set 7*35 on B to 7*5 on A, and over 7*5 on C read 57*71b. on A.
To find diameter in inches of a sphere of given weight.
Set the cursor to the given weight in lb. on A, and move
the slide until the same number is found on C under the cursor
that is simultaneously found on A over the constant for the
sphere on B.
Ex. — Find diameter in inches of a sphere of cast-iron to weigh 7 Mb.
Setting the cursor to 7*5 on A, and moving the slide, it is found
that when 3'8 on C falls under the cursor, 3*8 on A is simultaneously
found over 7*35 on B. The required diameter is therefore 3'8in.
The rules for cubes and cube roots (page 40) should be kept in
view in solving the last two examples.
A PRACTICAL MANUAL 61
Falling Bodies.
To find velocity in feet per second of a falling body, given the
time of fall in seconds.
Set index on C to time of fall on D, and under 32*2 on C read
velocity in feet per second on D.
To find velocity in feet per second, given distance fallen
through in feet.
Set 1 on C to distance fallen through on A, and under 64*4 on
B read velocity in feet per second on D.
Ex. — Find velocity acquired by falling through 14ft.
Set (k.h.) index of C to 14 on A, and under 64 "4 on B read 30ft.
per second on D.
To find distance fallen through in feet in a given time.
Set index of C to time in seconds on D, and over 16*1 on B
read distance fallen through in feet on A.
Centrifugal Force.
To find the centrifugal force of a revolving mass in lb.
Set 2940 on B to revolutions per minute on T> ; bring cursor
to weight in lb. on B ; index of B to cursor, and over radius in
feet on B read centrifugal force in lb. on A.
To find the centrifugal stress in lb. per square inch, in rims of
revolving wheels of cast iron.
Set 61*3 on C to the mean diameter of the wheel in feet on D,
and over revolutions per minute on C read stress per square inch
on A.
Ex. — Find the stress per square inch in a cast-iron fly-wheel rim 8ft.
in diameter and running at 120 revolutions per minute.
Set 61 '3 on C to 8 on D, and over 120 on C read 2451b. per square
inch on A.
The Steam Engine.
Given the stroke and number of revolutions per minute, to find
the piston speed.
Set stroke in inches on C to 6 on D, and over number of
revolutions on D read piston speed in feet per minute on C.
To find cubic feet of steam in a cylinder at cut-off, given
diameter of cylinder and period of admission in inches.
Set 2200 on B to cylinder diameter on D, and over period of
admission on B read cubic feet of steam on A.
62 THE SLIDE RULE I
Ex. — Cylinder diameter 26in., stroke 40in., cut-off at g of stroke.
Find cubic feet of steam used (theoretically) per stroke.
Set 2200 on B to 26 on D, and over 40 x § or 25in. on B, read
7*68 cub. ft. on A, as the number of cubic feet of steam used per
stroke.
Given the diameter of a cylinder in inches, and the pressure in
lb. per square inch, to find the load on the piston in tons.
Set pressure in lb. per square inch on B to 2852 on A, and over
cylinder diameter in inches on D read load on piston in tons on B.
Ex. — Steam pressure 1801b. per square inch ; cylinder diameter, 42in.
Find load in tons on piston.
Set 180 on B to 2852 on A, and over 42 on D read 111 tons, the
gross load, on B.
Given admission period and absolute initial pressure of steam
in a cylinder, to find the pressure at various points in the expansion
period (isothermal expansion).
Invert the slide and set the admission period, in inches, on O
to the initial pressure on D ; then under any point in the
expansion stroke on O find the corresponding pressure on D.
Ex. — Admission period 12in., stroke 42in., initial pressure 801b. per
square inch. Find pressure at successive fifths of the expansion period.
Set 12 on O to 80 on D, and opposite 18, 24, 30, 36 and 42in. of
the whole stroke on O find the corresponding pressures on D : — 53*3,
40, 32, 26 6 and 22*81b. per square inch.
To find the mean pressure constant for isothermally expanding
steam, given the cut-off as a fraction of the stroke.
Find the logarithm of the ratio of the expansion r, by the
method previously explained (page 46). Prefix the characteristic
and to the number thus obtained, on D, set 1 on C. Then under
2 '302 on C read x on D. To a?+l on D set r on C, and under
index of C read mean pressure constant on D. "The latter,
multiplied by the initial pressure, gives the mean forward pressure
throughout the stroke. (N.B.— Common log. x 2*302= hyperbolic
log.) '
Ex. — Find the mean pressure constant for a cut-off of £th, or a ratio
of expansion of 4.
Set (l.h.) index of C to 4 on D, and on the reverse side of the slide
read 0*602 on the logarithmic scale. The characteristic = ; hence to
0*602 on D set (k.h.) index of C, and under 2*302 on C read 1*384 on
]). Add 1, and to 2'384 thus obtained on D set r( = 4)on 0, and
under 1 on C read 0*596, the mean pressure constant required.
A PRACTICAL MANUAL 63
Mean pressure constants for the most usual degrees of cut-off
are given below : —
Cut-off in fractions of stroke... £ $. } % f |
Mean pressure constant 0*968 0*952 0-934 0*919 0*913 0*846
Cut-off in fractions of stroke... {• 4 ^ A J i
Mean pressure constant 0*766 0*750 0*699 0*664 0*596 0*522
Cut-off in fractions of stroke... % ±- \ 4 To tV
Mean pressure constant 0*465 0*421 0*385 0*355 0*330 0*309
Cut-off in fractions of stroke... ^V tV tV tV tV
Mean pressure constant 0*290 0*274 0*260 0*247 0*236
To find mean pressure : — Set 1 on C to constant on D, and
under initial pressure on C read mean pressure on D.
Given the absolute initial pressure, length of stroke, and
admission period, to find the absolute pressure at any point in
the expansion period, it being assumed that the steam expands
p
adiabatically. (P 2 =— join which P x = initial pressure and P 2
R 9
the pressure corresponding to a ratio of expansion R.)
Set l.h. index of C to ratio of expansion on D, and read on the
back of the slide the decimal of the logarithm. Add the charac-
teristic, and to the number thus obtained on D set 9 on C, and
read off the value found on D under the index of C. Set this
number on the logarithmic scale to the index mark, in the opening
on the back of the rule, and under l.h. index of C read the value
of R 1 ^ on D. The initial pressure divided by this value gives the
corresponding pressure due to the expansion.
Ex. — Absolute initial pressure 1201b. per square inch; stroke, 4ft.;
cut-off £. Find the respective pressures when ^ and f ths of the stroke
have been completed.
In the first case R = 2. Therefore setting the l.h. index of C to 2
on D, we find the decimal of the logarithm on the back of the slide to
be 0*301. The characteristic is 0, so placing 9 on C to 0'301 on D, we
read 0*334 as the value under the r.h. index of C. (N.B. — In
locating the decimal point it is to be observed that the log. of R has
been multiplied by 10, in accordance with the terms of the above
expression.) Setting this number on the logarithmic scale to the back
index, the value of R ^ is found on D, under the L. H. index of C, to
be 2*16. Setting 120 on C to this value, it is found that the pressure
at ^ stroke, read on C over the R.H. index of D, is 55*51b. per square
inch. In a similar manner, the pressure when § ths of the stroke is
completed is found to be 35*41b. per square inch.
For other conditions of expanding steam, or for gas or air, the
method of procedure is similar to the above.
64 THE SLIDE RULE:
To find the horse-power of an engine, having given the mean
effective pressure, the cylinder diameter, stroke, and number of
revolutions per minute.
To cylinder diameter on D set 145 on C ; bring cursor to stroke
in feet on B, 1 on B to cursor, cursor to number of revolutions on
B, 1 on B to cursor, and over mean effective pressure on B find
horse-power on A.
(N.B.— If stroke is in inches, use 502 in place of 145 given
above.)
Ex. — Find the indicated horse-power, given cylinder diameter 27in.,
mean effective pressure 381b. per square inch, stroke 32in. , revolutions 57
per minute.
Set 502 on C to 27 on D, bring cursor to 32 on B, 1 on B to cursor,
cursor to 57 on B, 1 on B to cursor, and over 38 on B read 200 I.H.P.
on A.
In determining the horse power of compound, triple, or quad-
ruple-expansion engines, invert the slide and set the diameter of
the high-pressure cylinder on to the cut-off in that cylinder on
A. Use the number then found on A over the diameter of the
fow-pressure cylinder on as the cut-off in that cylinder, working
with the same pressure and piston speed.
To find the cylinder ratio in compound engines^ invert the
slide and set index of to diameter of the low-pressure cylinder
on D. Then over the diameter of the high-pressure cylinder on
C, read cylinder ratio on A.
Ex. — Diameter of high-pressure cylinder 7|in. , low-pressure 15in.
Find cylinder ratio.
Set index on to 15 on D, and over 7*75 on read 3*75, the
required ratio, on A.
The cylinder ratios of triple or quadruple-expansion engines
may be similarly determined.
Ex. — In a quadruple-expansion engine, the cylinders are 18, 26. 37,
and 54 inches in diameter. Find the respective ratios of the high, first
intermediate, and second intermediate cylinders to the low-pressure.
Set (r.h.) index of to 54 on D, and over 18, 26, and 37 on read
9, 4'31, and 2*13, the required ratios, on A.
Given the mean effective pressures in lb. per square inch in
each of the three cylinders of a triple-expansion engine, the I.H.P.
to be developed in each cylinder, and the piston speed, to find the
respective cylinder diameters.
A PRACTICAL MANUAL 65
Set 42,000 on B to piston speed on A ; bring cursor to mean
effective pressure in low-pressure cylinder on B, index of B to
cursor, and under I.H.P. on A read low-pressure cylinder diameter
on C. To find the diameters of the high-pressure and inter-
mediate-pressure cylinders, invert the slide and place the mean
pressure in the low-pressure cylinder on £[ to the diameter of that
cylinder on D. Then under the respective mean pressures on 9
read corresponding cylinder diameters on D.
Ex. — The mean effective pressures in the cylinders of a triple-expan-
sion engine are :— L.P., 10-32 ; I. M.P., 27*5 ; and H.P. , 77 "51b. per square
inch. The piston speed is 650ft. per minute, and the I.H.P. developed
in each cylinder, 750. Find the cylinder diameters.
Set 42,000 on B to 650 on A, and bring cursor to 10'32 on B.
Bring index of B to cursor, and under 750 on A read 68*5in. on C, the
L.P. cylinder diameter. Invert the slide, and placing 10*32 on 8! to
68*5 on D, read, under 27*5 on 3, the I.M. P. cylinder diameter
= 42in., on D ; also under 77*5 on 3 read the H.P. cylinder diameter
= 25in., on D.
To compute brake or dynamometrical horse-power.
Set 525 on Cto the total weight in lb. acting at the end of the
lever (or pull of spring balance in lb.) on D ; set cursor to length
of lever in feet on C, bring 1 on C to cursor, and under number of
revolutions per minute on C find brake horse-power on D.
Given cylinder diameter and piston speed in feet per minute,
to find diameter of steam pipe, assuming the maximum velocity of
the steam to be 6000 ft. per minute.
Set 6000 on B to cylinder diameter on D, and under piston
speed on B read steam pipe diameter on D.
Given the number of revolutions per minute of a Watt
governor, to find the vertical height in inches, from the plane of
revolution of the balls to the point of suspension.
Set revolutions per minute on C to 35,200 on A, and over index
of B read height on A.
Given the weight in lb. of the rim of a cast-iron fly-wheel, to
find the sectional area of the rim in square inches.
Set the mean diameter of the wheel in feet on C to 0*102 on D,
and under weight of rim on C find area on D.
Given the consumption of coal in tons per week of 56 hours,
and the I.H.P., to find the coal consumed per I.H.P. per hour.
Set I.H.P. on C to 40 on D, and under weekly consumption on
C read lb. of coal per I.H.P., per hour on D.
66 THE SLIDE RULE:
Ex. — Find coal used per I. H.P. per hour, when 24 tons is the weekly
consumption for 300 I. H.P.
Set 300 on to 40 on D, and under 24 on read 3 '21b. per I. H.P.
per hour on D.
(N.B. — For any other number of working hours per week
divide 2240 by the number of working hours, and use the quotient
in place of 40 as above.)
To find the tractive force of a locomotive.
Set diameter of driving wheel in inches on B to diameter of
cylinder in inches on D, and over the stroke in inches on B read
on A, tractive force in lb. for each lb. of effective pressure on the
piston.
Steam Boilers.
To find the bursting pressure of a cylindrical boiler shell,
having given the diameter of shell and the thickness and ultimate
strength of the material.
Set the diameter of the shell in inches on C to twice the thick-
ness of the plate on D, and under strength of material per square
inch on C read bursting pressure in lb. per square inch on D.
Ex. — Find the bursting pressure of a cylindrical boiler shell 7ft. 6in.
in diameter, with plates £-in. thick, assuming an ultimate strength of
50,0001b. per square inch.
Set 90 on C to 1*0 on D, and under 50,000 on C find 5551b. on D.
To find working pressure for Fox's corrugated furnaces by
Board of Trade rule.
Set the least outside diameter in inches on C to 14,000 on D,
and under thickness in inches on C read working pressure on D in
lb. per square inch.
To find diameter d in inches, of round steel for safety valve
springs by Board of Trade rule.
Set 8000 on C to load on spring in lb. on D, and under the
mean diameter of the spring in inches on C read d 3 on D. Then
extract the cube root as per rule.
SrEED Ratios of Pulleys, Etc.
Given the diameter of a pulley and its number of revolutions
per minute, to find the circumferential velocity of the pulley or
the speed of ropes, belts, etc., driven thereby.
Set diameter of pulley in inches on C to 3*82 on D, and over
revolutions per minute on T) read speed in feet per minute on C.
A PRACTICAL MANUAL 67
Ex. — Find the speed of a belt driven by a pulley 53in. in diameter
and running at 180 revolutions per minute.
Set 53 on to 3*82 on D, and over 180 on D read 2500ft. per
minute on C.
Ex. — Find the speed of the pitch line of a spur wheel 3ft. 6in. in
diameter running at 60 revolutions per minute.
Set 42 in. on C to 3*82 on D, and over 60 on D read 660ft. per
minute on C.
Given diameter and number of revolutions per minute of a
driving pulley, and the diameter of the driven pulley, to find the
number of revolutions of the latter.
Invert the slide and set diameter of driving pulley on to
given number of its revolutions on D ; then opposite diameter of
any driven pulley on read its number of revolutions on D.
Ex. — Diameter of driving pulley 10ft.; revolutions per minute 55;
diameter of driven pulley 2ft. 9in. Find number of revolutions per minute
of latter.
Set 10 on to 55 on D, and opposite 2*75 on read 200 revolutions
onD.
Belts and Ropes.
To find the ratio of tensions in the two sides of a belt, given
the coefficient of friction between belt and pulley ^ and the
number of degrees in the arc of contact (log. R = y~~ j .
Set 132 on C to the coefficient of friction on D, and read off the
value found on D under the number of degrees in the arc of con-
tact on C. Place this value on- the scale of equal parts on the
back of the slide, to the index mark in the aperture, and read the
required ratio on D under the l.h. index of C.
Ex. — Find the tension ratio in a belt, assuming a coefficient of friction
of 0*3 and an arc of contact of 120 degrees.
Set 132 on C to 0'3 on D, and under 120 on C read 0*273. Place
this on the scale to the index on the back of the rule, and under the
L.H. index C read 1*875 on D, the required ratio.
Given belt velocity and horse-power to be transmitted, to find
the requisite width of belt, taking the effective tension at 501b.
per inch of width.
Set G60 on C to velocity in feet per minute on D, and opposite
horse-power on D find width of belt in inches on O.
Given velocity and width of belt, to find horse-power trans-
mitted.
Set 660 on C to velocity on D, and under width on C find
horse-power transmitted on D.
68 THE SLIDE RULE:
(N.B. — For any other effective tension, instead of 660 u
gauge point : — 33,000 -f tension.)
Given speed and diameter of a cotton driving rope, to find
power transmitted, disregarding centrifugal action, and assuming
an effective working tension of 2001b. per square inch of rope.
Set 210 on B to 1*75 on D, and over speed in feet per minute
on B read horse-power on A.
Ex. — Find the power transmitted by a lfiu. rope running at 4000ft.
per minute.
Set 210 on B to 1«75 on D, and over 4000 on B read 58*3 horse-
power on A.
Find the "centrifugal tension" in the previous example, taking
the weight per foot of the rope as=r0'27d 2 .
Set 655 on C to the diameter, l*75in., on D, and over the speed,
4000ft on C, read centrifugal tension = 1141b. on A.
Spur Wheels.
Given diameter and pitch of a spur wheel, to find number of
teeth.
Set pitch on C to 7T (3'1416) on D, and under any diameter on
C read number of teeth on D.
Given diameter and number of teeth in a spur wheel, to find
the pitch.
Set diameter on C to number of teeth on D, and read pitch on
C opposite 3*1416 on D.
Given the distance between the centres of a pair of spur wheels
and the number of revolutions of each, to determine their
diameters.
To twice the distance between the centres on D, set the sum of
the number of revolutions on C, and under the revolutions of each
wheel on C find the respective wheel diameters on D.
Ex. — The distance between the centres of two spur wheels is 37 "5in.,
and they are required to make 21 and 24 revolutions in the same time.
Find their respective diameters.
Set 21 + 24 = 45 on C to 75 (or 37 '5 x 2) on D, and under 21 and 24
on C find 35 and 40in. on D as the respective diameters.
To find the power transmitted by toothed wheels, given
the pitch diameter d in inches, the number of revolutions
per minute n, and the pitch p in inches, by the rule, H.P.
_nd p 2
m 400
A PRACTICAL MANUAL 69
Set 400 on B to pitch in inches on D ; set cursor to d on
B, 1 on B to cursor, and over any number of revolutions n on B
read power transmitted on A.
Ex. — Find the horse-power capable of being transmitted by a spur
wheel 7ft. in diameter, 3in. pitch, and running at 90 revolutions per
minute.
Set 400 on B to 3 on D ; bring cursor to 84in. on B, 1 on B to
cursor, and over 90 revolutions on B read 170, the horse -power trans-
mitted, on A.
Screw Cutting.
Given the number of threads per inch in the guide screw, to
find the wheels to cut a screw of given pitch.
Set threads per inch in guide screw on C, to the number of
threads per inch to be cut on D. Then opposite any number
of teeth in the wheel on the mandrel on C, is the number of
teeth in the wheel to be placed on the guide screw on D.
Strength of Shafting.
Given the diameter d of a steel shaft, and the number of
revolutions per minute n, to find the horse-power from : —
H.P.=d 3 x tixO-02.
Set 1 on C to d on D, and bring cursor to d on B. Bring 50
on B to cursor, and over number of revolutions on B read H.P.
on A.
Ex. — Find horse-power transmitted by a 3in. steel shaft at a 110
revolutions per minute.
Set 1 on C to 3 on D, and bring cursor to 3 on B. Bring 50 on B
to cursor, and over 110 on B read 59 '4 horse-power on A.
Given the horse-power to be transmitted and the number of
revolutions of a steel shaft, to find the diameter.
Set revolutions on B to horse-power on A, and bring cursor to
50 on B. Then move the slide until the same number is found on
B under the cursor that is simultaneously found on D under the
index of C., This number is the diameter, required.
To find the deflection h in inches, of a round steel shaft of
diameter d, under a uniformly distributed load in lb. w, and
supported by bearings, the centres of w T hich are I feet apart
( k = »*» \
\ K 78,000 d A J
Modifying the form of this expression slightly, we proceed as
follows : — Set o^onC to I on D, and bring the cursor to the same
70 THE SLIDE RULE:
number on B that is found on D under the index of C. Bring d
on B to cursor, cursor to w on B, 78,800 on B to cursor, and read
deflection on A over index of B.
Ex. — Find the deflection in inches of a round steel shaft 3|in.
diameter, carrying a uniformly distributed load of 3200 lb. , the distance
apart of the centres of support being 9ft.
Set 3*5 on C to 9 on D, and read 2*57 on D, under the l.h.
index of C. Set cursor to 2*57 on B, and bring 3*5 on B to cursor,
cursor to 3200 on B, 78,800 on B to cursor, and over l.h. index of B
read 0*197in., the required deflection on A.
To find the diameter of a shaft subject to twisting only, given
the twisting moment in inch-lb. and the allowable stress in lb. per
square inch.
Set the stress in lb. per square inch on B to the twisting
moment in inch-lb. on A, and bring cursor to 5*1 on B. Then
move the slide until the same number is found on B under the
cursor that is simultaneously found on D under the index of C.
Ex. — A steel shaft is subjected to a twisting moment of 2,700,000
inch-lb. Determine the diameter if the allowable stress is taken at 9000
lb. per square inch.
Set 9000 on B to 2,700,000 on A, and bring the cursor to 51 on
B. Moving the slide to the left, it is found that when 11*51 on the
R.H. scale of B is under the cursor, the l.h. index of C is opposite
11 '51 on D. This, then, is the required diameter of the shaft.
(N.B. — The rules for the scales to be used in finding the cube
root (page 42) must be carefully observed in working these
examples.)
Moments of Inertia.
To find the moment of inertia of a square section about an axis
formed by one of its diagonals ( 1= — ).
Set index of C to the length of the side of square s on D ; bring
cursor to s on C, 12 on B to cursor, and over index of B read
moment of inertia on A.
To find the moment of inertia of a rectangular section about an
axis parallel to one side and perpendicular to the plane of bending.
Set index of C to the height or depth h of the section, and
bring cursor to h on B. Set 12 on B to cursor, and over breach li
b of' the section on B read moment of inertia on A.
Ex. — Find the moment of inertia of a rectangular section of which
h = 14 in. and b = 7 in.
Set index of C to 14 on D, and cursor to 14 on B. Bring 12 on B
to cursor, and over 7 on B read 1600 on A.
A PRACTICAL MANUAL 71
Discharge from Pumps, Pipes, Etc.
To find the theoretical delivery of pumps, in gallons per
stroke.
Set 29'4 on B to the diameter of the plunger in inches on D,
and over length of stroke in feet on B read theoretical delivery in
gallons per stroke on A.
(N.B. — A deduction of from 20 to 40 per cent, should be made
to allow for slip.)
To find loss of head of water in feet due to friction in pipes
(Prony's rule).
Set diameter of pipe in feet on B to velocity of water in feet
per second on D and bring cursor to 2*25 on B ; bring 1 on B to
cursor, and over length of pipe in miles on B, read loss of head of
water in feet, on A.
To find velocity in feet per second, of water in pipes
(Blackwell's rule).
Set 23 on B to diameter of pipe in feet on A, and under
inclination of pipe in feet per mile on B read velocity in feet per
second on D.
To find the discharge over weirs in cubic feet per minute and
per foot of width. (Discharge = 214 \M 3 )*
Set 0*00467 on.C to the head in feet h on D, and under h on B
read discharge on D.
To find the theoretical velocity of water flowing under a given
head in feet.
Set index of B to head in feet on A, and under 64*4 on B read
theoretical velocity in feet per second on D.
HORSE-POWER OF WATER WHEELS.
To find the effective horse-power of a Poncelet water wheel.
Set 880 on C to cubic feet of flow of water per minute on D,
and under height of fall in feet on C, read effective horse-power
onD.
For breast water wheels use 960, and for overshot wheels 775,
in place of 880 as above.
Electrical Engineering.
To find the resistance per mile, in ohms, of copper svire of
high conductivity, at 60° F. the diameter being given in mils.
(1 mil. = 0-001in.).
72 THE SLIDE RULE:
Set diameter of wire in mils, on C to 54,900 on A, and over
r.h. or l.h. index of B read resistance in ohms on A.
Ex. — Find the resistance per mile of a copper wire 64 mils, in diameter,
Set 64 on C to 54,900 on A, and over r.h. index of B read 13*4
ohms on A.
To find the weight of copper wire in lb. per mile.
Set 7*91 on C to diameter of wire in mils, on D, and over index
of B read weight per mile on A.
Given electromotive force and current, to find electrical horse-
power.
Set 746 on C to electromotive force in volts on D, and under
current in amperes on C read electrical horse-power on D.
Given the resistance of a circuit in ohms and current in amperes,
to find the energy absorbed in horse-power.
Set 746 on B to current on D, and over resistance on B read
energy absorbed in H.P. on A.
Ex. — Find the H.P. expended in sending a current of 15 amperes
through a circuit of 220 ohms resistance.
Set 746 on B to 15 on D, and over 220 on B read 66 -3 H.P. on A.
Commercial.
To add on percentages.
Set 100 on C to 100+ given percentage on D, and under original
number on G read result on D.
To deduct percentages.
Set r.h. index of C to 100- the given percentage on D, and
under original number on C read result on D.
Ex. — From £16 deduct 1\ per cent.
Set 10 on 0, to 92*5 on D and under 16 on C, read 14*8 = £14, 16s.
on I). »
To calculate simple interest.
Set 1 on C to rate per cent, on D ; bring cursor to period
on C and 1 on C to cursor. Then opposite any sum on C find
simple interest on D.
For interest per annum.
Set r.h. index on C to rate on D, and opposite principal on C
read interest on D.
Ex. — Find the amount with simple interest of £250 at 8 per cent.,
and for a period of 1 year and 9 months.
Set 1 on C to 8 on I) ; bring cursor to 1 *75 on C, and 1 on C to
cursor : then opposite 250 on C read £35, the interest, on D. Then
250 + 35 = £285 = the amount.
A PRACTICAL MANUAL 73
To calculate compound interest.
Set the l.h. index of C to the amount of £1 at the given rate
of interest on D, and find the logarithm of this by reading on the
reverse side of the rule, as explained on page 46. Multiply the
logarithm, so found, by the period, and set the result, on the scale
of equal parts, to the index on the under-side of the rule ; then
opposite any sum on C read the amount (including compound
interest) on D.
Ex. — Find the amount of £500 at 5 per cent, for 6 years, with com-
pound interest.
Set L.H. index of C to £1*05 on D, and read at the index on the
scale of equal parts on the under side of rule, 0'0212. Multiply by 6,
we obtain 0'1272, which, on the scale of equal parts, is placed to the
index in the notch at the end of the rule. Then opposite 500 on C
read £670 on D, the amount required, including compound interest.
Miscellaneous Calculations.
To calculate percentages of compositions.
Set weight (or volume) of sample on C, to weight (or volume)
of substance considered, on D ; then under index of C read
required percentage on D.
Ex. — A sample of coal weighing l*25grms. contains 0'04425grm. of
ash. Find the percentage of ash.
Set 1-25 on O to 0*04425 on D, and under index on O read 3 '54,
the required percentage of ash on D.
Given the steam pressure P and the diameter d in millimetres,
of the throat of an injector, to find the weight W, of water
delivered in lb. per hour from W= q.^qf. '
Set 0*505 on C to P on A ; bring cursor to d on C and index of
C to cursor. Then under d on C read delivery of water on D.
To find the pressure of wind per square foot, due to a given
velocity in miles per hour.
Set 1 on B to 2 on A, and over the velocity in miles per hour
on D read pressure in lb. per square foot on B.
To find the kinetic energy of a moving body.
Set 64*4 on B to velocity in feet per second on D, and over
weight of body in lb. on B read kinetic energy or accumulated
work in foot-lb. on A.
74 THE SLIDE RULE:
TRIGONOMETRICAL APPLICATIONS
Scales. — Not the least important feature of the modern slide
rule is the provision of the special scales on the under-side of the
slide, and by the use of which, in conjunction with the ordinary
scales on the rule, a large variety of trigonometrical computa-
tions may be readily performed.
Three scales will be found on the reverse or under-side of the
slide of the ordinary Gravet or Mannheim rule. One of these is
the evenly-divided scale or scale of equal parts referred to in
previous sections, and by which, as explained, the decimal parts or
mantissse of logarithms of numbers may be obtained. Usually
this scale is the centre one of the three, but in some rules it will
be found occupying the lowest position, in which case some little
modification of the following instructions will be necessary. The
requisite transpositions will, however, be evident when the
purposes of the scales are understood. The upper of the three
scales, usually distinguished by the letter S, is a scale giving the
logarithms of the sines of angles, and is used to determine the
natural sines of angles of from 35 minutes to 90 degrees. The
notation of this scale will be evident on inspection. The main
divisions 1, 2, 3, etc., represent the degrees of angles ; but the
values of the subdivisions differ according to their position on the
scale. Thus, if any primary space is subdivided into 12 parts, each
of the latter will be read as 5 minutes (5'), since 1°=60 / .
Sines of Angles. — To find the sine of an angle the slide is placed
in the groove, with the under-side uppermost, and the end division
lines or indices on the slide, coinciding with the right and left
indices of the A scale. Then over the given angle on S is
read the value of the sine of the angle on A. If the result is found
on the left scale of A (1 to 10), the logarithmic characteristic is
-2 ; if it is found on the right-hand side (10 to 100), it is- 1. In
other words, results on the right-hand scale are prefixed by the
decimal point only, while those on the left-hand scale are to be
preceded by a cypher also. Thus : —
Sine 2° 40' = 0'0465 ; sine 15° 40' = 0'270.
Multiplication and division of the sines of angles arc per-
formed in the same manner as ordinary calculations, excepting
A PRACTICAL MANUAL
75
that the slide has its under-face placed uppermost, as just explained.
Thus to multiply sine 15° 40' by 15, the r.h. index of S is brought
to 15 on A, and opposite 15° 40' on S is found 405 on A. Again,
to divide 142 by sine 16° 30', we place 16 u 30' on S to 142 on A,
and over r.h. index of S read 500 on A.
The rules for the number of integers in the results are thus
determined : Let N be the number of integers in the multiplier M
or in the dividend D. Then the number of integers P, in the
product or Q, in the quotient are as follows : —
When the result is found to the right of M or D,
and in the same scale
When the result is found to the right of M or D,
and in the other scale
When the result is found to the left of M or I),
and in the other scale
When the result is found to the left of M or D,
and in the same scale
p=
=N-
■»l
p=
=N-
-1
p.
= N-
-1
F=
= N
Q = N
Q = N + 1
Q = N + 1
Q = X + 2
If the division is of the form
20° 30'
"Tor'
the result cannot be
read off directly on the face of the rule. Thus, if in the above ex-
ample 20° 30' on S, is placed to agree with 50 on the right-hand scale
of A, the result found on S under the r.h. index of A is 44° 30'.
The required numerical value can then be found : (1) By placing
the slide with all indices coincident when opposite 44° 30' on S
will be found 0*007 on A ; or (2)' In the ordinary form of rule, by
reading off on the scale B opposite the index mark in the opening
on the under-side of the rule. The above rules for the number of
integers in the quotient do not apply in this case.
If it is required to find the sine of an angle simply, this may
be done with the slide in its ordinary position, with scale B under
A. The given angle on scale S is then set to the index on the
under-side of the rule, and the value of the sine is read off on B
under the right index of A.
Owing to the rapidly diminishing differences of the values of
the sines as the upper end of the scale is approached, the sines of
angles between 60° and 90° cannot be accurately determined in the
foregoing manner. It is therefore advisable to calculate the value
of the sine by means of the formula :
2
G
Sine = 1-2 sin 2 -
76 THE SLIDE RULE:
To determine the value of sin 2 — — . With the slide in the
normal position, set the value of — Z— on S to the index on the
under-side of the rule, and read off the value ^onB under the
r.h. index of A. Without moving the slide lind x on A, and read
under it on B the value required.
Ex. — Find value of sine 79° 40'.
Sine 79° 40' = 1 - 2 sin 2 5° 10'.
But sine 5° 10' = 0*0900, and under this value on A is 0*0081 on
B. Therefore sine 79° 40' = 1 - 0*0162 = 0*9838.
The sines of very small angles, being very nearly proportional
to the angles themselves, are found by direct reading. To
facilitate this, some rules are provided with two marks, one of
which, a single accent ('), corresponds to the logarithm of -j p
and is found at the number 3438. The other mark — a double
accent (") — corresponds to the logarithm of ■ • -,, / and is found at
the number 206,265. In some rules these marks are found on
either the A or the B scales ; sometimes they are on both.
In either case the angle on the one scale is placed so as to
coincide with the significant mark on the other, and the result
read off on the first-named scale opposite the index of the
second.
In sines of angles under 3", the number of integers in the
result is- 5; while it is -4 for angles from 3" to 21"; -3 from
21" to 3' 27" ; and -2 from 3' 27" to 34' 23".
Ex. — Find sine 6'.
Placing the significant mark for minutes coincident witli 6,
the value opposite the index is found to be 175, and by the rule
above this is to be read 0*00175. For angles in seconds the other
significant mark is used ; while angles expressed in minutes and
seconds are to be first reduced to seconds. Thus, 3' 10"= 190".
Tangents of Angles. — There remains to be considered the third
scale found on the back of the slide, and usually distinguished
from the others by being lettered T. In most of the more recent
forms of rule this scale is placed near the lower edge of the slide,
but in some arrangements it is found to be the centre scale of the
three. Again, in some rules this scale is figured in the same
A PRACTICAL MANUAL 77
direction as the scale of sines — viz., from left to right, — while in
others the T scale is reversed. In both cases there is now usually
an aperture formed in the back of the left extremity of the rule,
with an index mark similar to that already referred to in
connection with the scale of sines. Considering what has been
referred to as the more general arrangement, the method of
determining the tangents of angles may be thus explained : —
The tangent scale will be found to commence, in some rules, at
about 34', or, precisely, at the angle whose tangent is O'Ol. More
usually, however, the scale will be found to commence at about
5° 43', or at the angle whose tangent is 0*1. The other extremity
of the scale corresponds in all cases to 45°, or the angle
whose tangent is 1. This explanation will suggest the method
of using the scale, however it may be arranged. If the
graduations commence with 34', the T scale is to be used in con-
junction with the right and left scales of A ; while if they
commence with 5° 43' it is to be used in conjunction with the
D scale.
In the former case the slide is to be placed in the rule so that
the T scale is adjacent to the A scales, and, with the right and
left indices coinciding, when opposite any angle on T will be found
its tangent on A. From what has been said above, it follows that
the tangents read on the l.h. scale of A have values extending
from 0*01 to 1 ; while those read on the r.h. scale of A have
values from 0*1 to 1*0. Otherwise expressed, to the values of
any tangent read on the l.h. scale of A a cypher is to be pre-
fixed ; while if found on the r.h. scale, it is read directly as a
decimal.
Ex.— Find tan. 3° 50'.
Placing the slide as directed, the reading on A opposite 3° 50'
on T is found to be 67, As this is found on the l.h. scale of A,
it is to be read as 0*067.
Ex.— Find tan. 17° 45'.
Here the reading on A opposite 17° 45' on T is 32, and as it is
found on the r.h. scale of A it is read on 0*32.
As in the case of the scale of sines, the tangents may be found
without reversing the slide, when a fixed index is provided in the
back of the rule for the T scale.
We revert now to a consideration of those rules in which a
single tangent scale is provided. It will be understood that in this
78 THE SLIDE RULE:
case the slide is placed so that the scale T is adjacent to the D
scale, and that when the indices of both are placed in agreement,
the value of the tangent of any angle on T (from 5° 43' to 45°)
may be read off on D, the result so found being read as wholly
decimal. Thus tan. 13° 20' is read 0'237.
If a back index is provided, the slide is used in its normal
position, when, setting the angle on the tangent scale to this
index, the result can be read on over the l.h. index
of D.
The tangents of angles above 45° are obtained by the formula :
Tan. d= - ,. For all angles from 45° to (90° -5° 43')
tan. (90-0) ° v J
we proceed as follows: — Place (90-0) on T to the r.ii. index of
D, and read tan. $ on D under the l.h. index of T. The first
figure in the value thus obtained is to be read as an integer. Thus,
to find tan. 71° 20' we place 90° -71° 20' = 18° 40' on T. to the r.ii.
index of D, and under the l.h. index of T read 296, the required
tangent.
The tangents of angles less than 40' are sensibly proportional
to the angles themselves, and as they may therefore be considered
as sines, their value is determined by the aid of the single, and
double accent marks on the sine scale, as previously explained.
The rules for the number of integers are the same as for the
sines.
Multiplication and division of tangents may be quite readily
effected.
Ex.— Tan. 21° 50' x 15 = 6.
Set l.h. index of T to 15 on D, and under 21° 50' on T read 6
onD.
Ex.— Tan. 72° 40' x 117 = 375.
Set (90°-72° 40') = 17° 20' on T to 117 on D, and under r.ii.
index of T read 375 on D.
Cosines of Angles. — The cosines of angles may be determined by
placing the scale S with its indices coinciding with those of A, and
when opposite (90-0) on S is read cos. B on A. If the result is
read on the l.h. scale of A, a cypher is to be prefixed to the value
read ; while if it is read on the R.H. scale of A, the value is read
directly as a decimal. Thus, to determine cos. 86° 30' we find
opposite (90°-86 30') = 3° 30' on S, 61° on A, and as this is en the
I,. li. scale the result is read o oci. Again, to find COS. 59 -Jo' we
A PRACTICAL MANUAL 79
road opposite (90° - 59° 20') or 30° 40' on S, 51 on A, and as t'his is
found on the r.h. scale of A, it is read 0*51.
In finding the cosines of small angles it will be seen that direct
reading on the rule becomes impossible for angles of less than 20°.
It is advisable in such cases to adopt the method described for de-
termining the sines of the large angles of which the complements
are sought.
Cotangents of Angles. — From the methods of finding the tangents
of angles previously described, it will be -apparent that the cotan-
gents of angles may also be obtained with equal facility. For
angles between 5° 45' and 45°, the procedure is the same as that for
finding tangents of angles greater than 45°. Thus, the angle on
scale T is brought to the r.h. index of D, and the cotangent read
off on D under the l.h. index of T. The first figure of the result so
found is to be read as an integer.
If the angle (0) lies between 45° and 84° 16', the slide is placed
so that the indices of T coincide with those of D, and the result is
then read off on D opposite (90 - 6) on T. In this case the value is
wholly decimal.
Secants of Angles. — The secants of angles are readily found by
bringing (90 - 6) on S to the r.h. index of A and reading the result
on A over the l.h. index of S. If the value is found on the l.h.
scale of A, the first figure is to be read as an integer ; while if the
result is read on the r.h. scale of A, the first tivo figures are to be
regarded as integers.
Cosecants of Angles. — The cosecants of angles are found by
placing the angle on S to the r.h. index of A, and reading the
value found on A over the l.h. index of S. If the result is read
on the l.h. scale of A, the first figure is to be read as an integer ;
while if the result is found on the r.h. scale of A, the first two
figures are to be read as integers.
It will be noted that some of the rules here given for determin-
ing the several trigonometrical functions of angles apply only to
those forms of rules in which a single scale of tangents T is used,
reading from left to right. For the other arrangements of the
scale, previously referred to, some slight modification of the method
of procedure in finding the tangents and cotangents of angles will
be necessary ; but as in each case the nature and extent of this
modification is evident, no further directions are required.
80 THE SLIDE RULE:
THE SOLUTION OF EIGHT- ANGLED TRIANGLES.
From the foregoing explanation of the manner of determining the
trigonometrical functions of angles, the methods of solving right-
angled triangles will be readily perceived, and only a few examples
need therefore be given.
Let a and b represent the sides and c the hypothenuse of a
right-angled triangle, and a and b° the angles opposite to the sides.
Then of the possible cases we will take
(1.) Given c and a , to find a, 6, and b°.
The angle b° = 90 -a , while a — c sin a and b = c sin b°. To find
a, therefore, the index of S is set to c on A, and the value of a read
on A opposite a on S. In the same manner the value of b is
obtained.
Ex. — Given in a right-angled triangle c=9 ft. and a°=30°.
Find a, 6, and b°.
The angle 6° = 90 - 30 = 60°. To find a, set r.h. index of S to 9
on A, and over 30° on S read a = 4*5 ft. on A. Also, with the slide
in the same position, read 6=7*8 ft. [7794] on A over 60° on S.
(2.) Given a and c, to determine a , £>°, and b.
In this case advantage is taken of the fact that in every triangle
the sides are proportional to the sines of the opposite angles,
Therefore, as in this case the hypothenuse c subtends a right angle,
of which the sine = l, the r.h. index (or 90°) on S is set to the
length of c on A, when under a on A is found a on S. Hence b°
and b may be determined.
(3.) Given a and a , to find 6, c, and b°.
Here b°=(90-a°), and the solution is similar to the foregoing.
(4.) Given a and b, to find a , 6°, and c.
To find a , we have tan. a°= — , which in the above example
4*5
will be — =0"577. Therefore, placing the slide so that the indices
7 '8
of T coincide with those of D, we read opposite 0*577 on D the
value of a°=30°. The hypothenuse c is readily obtained from
A PRACTICAL MANUAL $1
THE SOLUTION OF OBLIQUE-ANGLED TRIANGLES.
Using the same letters as before to designate the three sides
and the subtending angles of oblique-angled triangles, we have
the following cases : —
(1.) Given one side and two angles, as a, a , and b°, to find b, c,
and c°.
In the first place, c o =180°-(a o + 6°) ; also we note that, as
the sides are proportional to the sines of the opposite angles,
, a sine b° -, a sine c°
b = —. and c— — .
sine a sine a°
Taking as an example, a = 45, a°=57°, and b°= 63°, we have c°=
180 -(57 + 63) = 60°. To find b and c, set a on S to a on A, and
read off on A above 63° and 60° the values of b ( = 47'8) and c
( = 46*4) respectively.
(2.) Given a, 6, and a , to find b°, c°, and c.
In this case the angle a on S is placed under the length of side
a on A and under b on A is found the angle b° on S. The angle
c°=lS0-(a° +6°), whence the length c can be read off on A over
c° on S.
(3.) Given the sides and the included angle, to find the other
side and the remaining angles.
If, for example, there are given a = 65, 5 = 42, and the included
angle c°=55°, we have (a + b): (a-b) = tan. : tan —
Then, since a + b° = 180° -55° =125°, it follows that ?L±*1 = 1??=
' 2 2
62° 30'.
By the rule for tangents of angles greater than 45°, we find
tan. 62° 30'= 1 92. Inserting in the above proportion the values
thus found, we have 107 : 23 = 1*92 : tan. ^—^1. From this it-
is found that the value of the tangent is 0'412, and placing the
slide with all indices coinciding, it is seen that this value on D
corresponds to an angle of 22° 25'. Therefore, since a = 62°
30', and ? ~ b ° =22° 25', it follows that a =84° 55', and b° = 40° 5'.
Z
Finally, to determine the side c, we have c= a . sin c as before.
sin a
THE .SLIDE RULE:
PKACTICAL TRIGONOMETRICAL APPLICATIONS.
A few examples illustrative of the application of the methods
of determining the functions of angles, etc., described in the
preceding section, will now be given.
To find the chord of an arc, having given the included angle
and the radius.
With the slide placed in the rule with the C and D scales out-
ward, bring one-half of the given angle on S to the index mark in
the back of the rule, and read the chord on B under twice the
radius on A.
Ex. — Required the chord of an arc of 15°, the radius being 23in.
Set 7° 30' on S to the index mark in the back of the rule, and under
46 on A read 6in. , the required length of chord on B.
To find the area of a triangle, given two sides and the included
angle.
Set the angle on S to the index mark on the back of the rule,
and bring cursor to 2 on B. Then bring the length of one side on
B to cursor, cursor to 1 on B, the length of the other side on B to
cursor, and read area on B under index of A.
Ex. — The sides of a triangle are 5 and 6ft. in length respectively, and
they include an angle of 20°. Find the area.
Set 20 on S to index mark, bring cursor to 2 on B, 5 on B to cursor,
cursor to 1 on B, 6 on B to cursor, and under 1 on A read the aiva —
5*13 sq. ft. on B.
To find the number of degrees in a gradient, given the rise per
cent.
Place the slide with the indices of T coincident with those of
D, and over the rate per cent, on D read number of degrees in the
slope on T.
As the arrangement of rule we have chiefly considered has only
a single T scale, it will be seen that only solutions of the above
problem involving slopes between 10 and 100 per cent, can be
directly read off. For smaller angles, one of the formulae for the
determination of the tangents of sub-multiple angles must be used.
In rules having a double T scale (which is used with the A
scale) the value in degrees of any slope from 1 to 100 per cent, can
be directly read off on A.
To find the number of degrees, when the gradient is expressed
jis 1 in x.
A PliACflCAL MANUAL 83
Place the index of T to x on D, and over index of D read the
required angle in degrees on T.
Ex. — Find the number of degrees in a gradient of 1 in 3*8.
Set 1 on T to 3 '8 on D, and over r.h. index of D read 14° 45'
onT.
Given the lap, the lead and the travel of an engine slide valve,
to find the angle of advance.
Set (lap + lead) on B to half the travel of the valve on A, and
read the angle of advance on S at the index mark on the back of
the rule.
Ex. — Valve travel 4|in., lap lin., lead r 5 gin. Find angle of advance.
Set 1^ = 1*312 on B to 2*25 on A, and read 35° 40' on S opposite
the index on the back of the rule.
Given the angular advance 0, the lap and the travel of a slide
valve, to find the cut-off in percentage of the stroke.
Place the lap on B to half the travel of valve on A, and read
on S the angle (the supplement of the angle of the eccentric) found
opposite the index in the back of the rule. To this angle, add the
angle of advance and deduct the sum from 180°, thus obtaining
the angle of the crank at the point of cut-off. To the cosine of the
supplement of this angle, add 1 and multiply the result by 50,
obtaining the percentage of stroke completed when cut- off occurs.
Ex. — Given the angular advance = 35° 40', the valve travel = 4^in.,
and the lap = lin., find the angle of the crank at cut-off and the admission
period expressed as a percentage of the stroke.
Set 1 on B to 2*25 on A, and read off on S opposite the
index, the supplement of the angle of the eccentric = 26° 20'. Then
180° -(35° 40' + 26° 20') = 118° = the crank angle at the point of cut-off.
Further, cos. 118° = cos. 62° = sin (90° -62°) = sin 28°, and placing 28°
on S to the back index, the cosine, read on B under R.H. index of
A, is found to be 0*469. Adding 1 and placing the L.H. index of C to
the result, 1*469, on D, we read off under 50 on C, the required period
of admission = 73 - 4 per cent, on D.
The trigonometrical scales are useful for evaluating certain
formulae. Thus in the following expressions, if we find the angle
a such that sin. a = k, we can write : —
h /-I 7.>
- =tan. a ; 2— — — = cot. a; *Jl-k 2 = cos. a; etc.
\f 1 — rC~ rC
In the tirst expression, take k= 0*298. Place the slide with the
sine scale outward and with its indices agreeing with the indices
of the rule. Set the cursor to 0*298 on the (r.h.) scale of A, and
read 17° 20' on the sine scale as the angle required. Then under
17° 20' on the tangent scale, read 0*312 on ~D as the result.
84 THE SLIDE RULE:
SLIDE RULES WITH LOG.-LOG. SCALES.
For occasional requirements, the method described on page 45
of determining powers and roots other than the square and cube,
is quite satisfactory. When, however, a number of such calcula-
tions are to be made, the process may be simplified considerably
• by the use of what are known as log.-log., logo-log., or logometric
scales, in conjunction with the ordinary scales of the rule. The
principle involved will be understood from a consideration of
those rules for logarithmic computation (page 8) which refer to
powers and roots. From these it is seen that while for the multi-
plication and division of numbers we add their logarithms, for
involution and evolution we require to multiply or divide the
logarithms of the numbers by the exponent of the power or root
as the case may be. Thus to find 3 2,3 , we have (log. 3) x 2*3 = log.
x, and by the ordinary method described on page 45 we should
determine log. 3 by the aid of the scale L on the back of the slide,
multiply this by 2 '3 by using the C and D scales in the usual
manner, transfer the result to scale L, and read the value of x on
D under 1 on C. By the simpler method, first proposed by Dr.
P. M. Roget,* the multiplication of log. 3 by 2*3 is effected in the
same way as with any two ordinary factors — i.e., by adding their
logarithms and finding the number corresponding to the resulting
logarithm. In this case we have log. (log. 3) + log. 2*3 = log. (log.
x). The first of the three terms is obviously the logarithm of the
logarithm of 3, the second is the simple logarithm of 2*3, and the
third the logarithm of the logarithm of the answer. Hence, if we
have a scale so graduated that the distances from the point of
origin represent the logarithms of the logarithms (the log.-logs.)
of the numbers engraved upon it, then by using this in conjunc-
tion with the ordinary scale of logarithms, we can effect the
required multiplication in a manner which is both expeditious
and convenient. Slightly varying arrangements of the log.-log.
scale, sometimes referred to as the " P line," have been introduced
from time to time, but latterly the increasing use of exponential
formulae in thermodynamic, electrical, and physical calculations
has led to a revival of interest in Dr. Roget's invention, and various
arrangements of rules with log.-log. scales are now available.
* Philosophical Transactions of the Royal Society, 1815.
A PRACTICAL MANUAL 85
The Davis Log. -Log. Rule. — In the rule introduced by Messrs.
John Davis & Son Limited, Derby, the log.-log. scales are placed
upon a separate slide — a plan which has the advantage of leaving
the rule intact for all ordinary purposes, while providing a length
of 40in. for the log.-log. scales.
In the lOin. Davis rule one face of the slide, marked E, has
two log.-log. scales for numbers greater than unity, the lower
extending from 1*07 to 2, and the upper continuing the gradua-
tions from 2 to 1000. On the reverse face of the slide, marked
- E, are two log.-log. scales for numbers less than unity, the upper
extending from 0001 to 0'5, and the lower continuing the gradua-
tions from 0*5 to 0'933. Both sets of scales are used in conjunc-
tion with the lower or D scale of the rule, which is to be primarily
regarded as running from 1 to 10, and constitutes a scale of
exponents. In the 20in. rule the log.-log. scales are more exten-
sive, and are used in conjunction with the upper or A scale of the
rule (1 to 100) ; in what follows, however, the lOin. rule is more
particularly referred to.
It has been explained that on the log.-log. scale the distance of
any numbered graduation from the point of origin represents
the log.-log. of the number. The point of origin will obviously
be that graduation whose log.-log. = 0. This is seen to be 10, since
log. (log. 10) = log. 1=0. Hence, confining attention to the E scale,
to locate the graduation 20, we have log. (log. 20) = log. 1'301 =
0'11397, so that if the scale D is 25cm. long, the distance between
10 and 20 on the corresponding log.-log. scale would be 113'97-^4
= 28*49nim. For numbers less than 10 the resulting log.-logs. will
be negative, and the distances will be spaced off from the point
of origin in a negative direction — i.e., from right to left. Thus, to
locate the graduation 5, we have
log. (log. 5) = log. 0-699 = 1-844; i.e., - 1 +0'844 or-0'156 ;
so that the graduation marked 5 would be placed 156 -^ 4 = 39 mm.
distant from 10 in a negative direction, and proceeding in a similar
manner, the scale may be extended in either direction. In the
- E scale, the notation runs in the reverse direction to that of the E
scale, but in all other respects it is precisely analogous, the distance
from the point of origin (0'1 in this case) to any graduation x
representing log. [-log. #.]. It follows that of the similarly
situated graduations on the two scales, those on the - E scale are
the reciprocals of those on the E scale. This may be readily verified
86 THE SLIDE RULE:
by Betting, say, 10 on E to (r.h.) 1 on D, when turning to the hack
of the rale we find O'l on -E agreeing with the index mark in the
aperture at the right-hand extremity of the rule.
In using the log.-log. scales it is important to observe (1) that
the values engraved on the scale are definite and unalterable («.</.,
1*2 can only be read as 1*2 and not as 120, 0*0012, etc., as with the
ordinary scales) ; (2) that the upper portion of each scale should
be regarded as forming a prolongation to the right of the lower
portion ; and (3) that immediately above any value on the lower
portion of the scale is found the 10th power of that value on the
upper portion of the scale. Keeping these points in view, if we
set 1*1 on E to 1 on D we find over 2 on D the value of 1*1--- 1*21
on E. Similarly, over 3 we find 1*1 3 =1'331, and so on. Then,,
reading across the slide, we have, over 2, the value of l'l*x*=l -120
= 6'73, and over 3 we have li»x»ss'Vl»«17'4, Hence the rule :—
To find the value of x n , set x on E to 1 on 2>, and over n on D read
xn on E.
With the slide set as above, the 8th, 9th, etc., powers of 1*1
cannot be read off ; but it is seen that, according to (2) in the
foregoing, the missing portion of the E scale is that part of the
upper scale (2 to about 2*6) which is outside the rule to the left.
Hence placing 1*1 to 10 on D, the 8th, 9th, etc., powers of l'l will
be read off on the upper part of the E scale. In general, then,
If x on the lower line is set to 1 on D, then x n is read directly
on that line and x 10n on the upper line.
If x on the upper line is set to 1 on D, then x n is read directly
n
on that line and x 115 on the lower line.
n
If x on the lower line is set to 10 on D, then x 115 is read
directly on that line and x n on the upper line.
_n
If x on the upper line i3 set to 10 on D, then x lTi is read
_ w_
directly on that line and x iT)0 on the lower line.
These rules are conveniently exhibited in the accompanying
diagram (Fig. 14). They are equally applicable to both the E and
-E scales of the lOin. rule, and include practically all the
instruction required for determining the nth power or the ftth
root of a number. They do not apply directly to the 20in. rule,
however, for here the relation of the lower and upper scales will
be xn and x m n.
A PRACTICAL MANUAL
87
Ex.— Find 1-1 67 200 .
Set 1 '167 on E to 1 on D, and over 2*56 on 1) read 1 '485 on E.
Ex.— Find4-G M!1 .
Set 4'G on upper E scale to 1 on D, and over 1*61 on D read 11*7
(11-67) on E.
Ex.— Find l-4°--7 and V4?' r .
Set 1 -4 en E to 10 on D, and over 2 "7 on D read 1-095 = lM *- 7 on
lower E scale and 2*48 = 1 '4'" 7 on upper E scale.
l x IOrt
10
hT
10
\x n
E
i
ir'o
\X
D
n
1
10
1 a
\x
•a? to
E
I
I 1
n
1
10
D
Fig. 14.
Ex.— Find 46 ' 0184 and 46°- 184 .
Set 46 on upper E scale to 10 on D, and over 1*84 on D read 1'073
on lower E scale and 2'022 (2 '0223) on upper E scale.
Ex. — FindO-074 1 * 15 .
Using the-E scale, set 0*074 to 1 on D, and over 1*15 on D read
0'05 on-E.
The method of determining the root of a number will be
obvious from the preceding examples.
Ex.— Find i\JTl and U JT7.
Set 17 <>n E to 1 -4 on D, and over 1 on D read 7 '56 on upper E
scale and 1 '224 on lower E scale.
88 THE SLIDE RULE:
Ex.— Find °' 08 i/0 : 9l4.
Set 0-914 on-E to 3'1 on D, and over 10 on D read 0*055 on
upper - E scale.
When the exponent n is fractional, it is often possible to
obtain the result directly with one setting of the slide. Thus to
17
determine 1*135 16 by the first method we find |$- = 1*0625, and
placing T135 on E to 1 on D, read 1*144 on E over 1*0625 on D.
By the direct method we place 1*135 on the E scale on 1*6 on D,
and over 1*7 on D read 1*144 on E. It will be seen that since the
scale D is assumed to run from 1 to 10 we are unable to read 16
and 17 on this scale ; but it is obvious that the ratios )tg and J-J are
identical, and it is with the ratio only that we are, in effect,
concerned,
Since an expression of the form x" n = L or ( ) , the required
x n \xj
value may be obtained by first determining the reciprocal of x
and proceeding as before. By using both the direct and reciprocal
log.-log. scales (E and-E) in conjunction however, the required
value can be read directly from the rule, and the preliminary
calculation entirely avoided. In the Davis form of rule, the result
can be read on the-E scale, used in conjunction with the D scale
of the rule, x on E being set to the index mark in the aperture in
the back of the rule.
Ex.— Find the value of 1-195- 1 * 65 .
Set 1*195 on E to the index in the left aperture in the back of the
rule, and over 1*65 on D read 0*745 on the-E scale.
It may be noted in passing that the log.-log. scale affords a
simple means for determining the logarithm or anti-logarithm of
a number to any base. For this purpose it is necessary to set the
base of the given system on E to 1 on D, when under any number
on E will be found its logarithm on D. Thus, for common logs.,
we set the base 10 on E to 1 on D, and under 100 we find 2, the
required log. Similarly we read log. 20 = 1*301 ; log. 55 = 1*74;
log. 550 = 2*74, etc. Reading reversely, over 1*38 on D we find its
antilog. 24 on E ; also antilog. 1*58 = 38 ; antilog. 1*19 = 15*5, etc.
For logs, of numbers under 10 we set the base 10 to 10 on D ;
hence the readings on D will be read as one-tenth their apparent
value. Thus log. 3=0*477; log. 5*25 = 0*72; antilog. 0*415 = 2*6;
antilog. 0*525 = 3.35, etc.
A PRACTICAL MANUAL 89
The logs, of the numbers on the lower half of the E scale
will also be found on the D scale ; but a consideration of
Fig. 14 will show that this will be read as one-tenth its face
value if the base is set to 1 on D, and as one-hundredth if the
base is set to 10.
For natural, hyperbolic, or Napierian logarithms, the base is
2*718. A special line marked e or e serves to locate the exact
position of this value on the E scale, and placing this to 1 on D
we read log.* 4'35 = 1'47; lo^.e 7'4 = 2'0; antilog.ex2'89 = 18, etc.
The other parts of the scale are read as already described for
common logs. Calculations involving powers of e are frequently
met with, and these are facilitated by using the special graduation
line referred to, as will be readily understood.
If it is required to determine the power or root of a number
which does not appear on either of the log. -log. scales, we may
break up the number into factors. Usually it is convenient to
make one of the factors a power of 10.
Ex.— 3950 1 ' 97 = 3'95 1 - 97 x 10 3 * 1 97 = 3-951.97 x 105 .9i #
Then 3'95 1,97 =15, and 10 5 ' 91 (or antilog.) 5*91 = 812,000. Hence,
15 x 812,000 = 12,180,000 is the result sought.
Numbers which are to be found in the higher part of the log.-
log. scale may often be factorised in this way, and greater accuracy
obtained than by direct reading.
The form of log.-log. rule which has been mainly dealt with in
the foregoing gives a scale of comparatively long range, and the
only objection to the arrangement adopted is the use of a separate
slide.
The Jackson-Davis Double Slide Rule. — In this instrument a
pair of aluminium clips enable the log.-log. slide to be temporarily
attached to the lower edge of the ordinary rule, and used, by
means of a special cursor, in conjunction with the C scale of the
ordinary slide. In this way both the log.-log. and ordinary scales
are available without the trouble of replacing one slide by the
other. Since the scale of exponents is now on the slide, the value
of x n will be obtained by setting 1 on C to x on E and reading the
result on E under n on C.
By using a pair of log.-log. slides, one in the rule and one
clamped to the edge by the clips, we have an arrangement which
is very useful ill deducing empirical formulae of the type yzzx 71 .
90
THE SLIDE RULE:
The Yokota Slide Rule. — In this instrument the log.-log. scales
are placed on the face of the rule, each set comprising three lines.
These, for numbers greater than 1, are found above the A scale
while the three reciprocal log.-log. lines are below the D scale.
Both sets are used in conjunction with the C scale on the slide.
Other features of this rule are : — The ordinary scales are lOin. long
Fig. 15.
instead of 25cm. as hitherto usual ; hence the logarithms of num-
bers can be read on the ordinary scale of inches on the edge of
the rule. There is a scale of cubes in the centre of the slide and
on the back of the slide there is a scale of secants in addition to
the sine and tangent scales.
The Faber Log.-log. Rule. — In this instrument shown in Fig. 15,
the log.-log. scales are placed on the bevelled edge, which formerly
\t log
► log.« =
— a
~r
100
10
D
Fig. 16.
carried a measuring scale of centimetres. Projecting from the
cursor and extending over the bevelled face is a metal tongue, the
end of which forms an index: or marker coinciding with the line
on the cursor glass. The two Log.-log. scales are arranged ride by
side, one extending from 1*1 to 2"9 and the other from iM) to
100,000 ; they are used in conjunction with the ( ! Bcale of the slide
in the manner previously described,
A PRACTICAL MANUAL 91
Another novel feature of this rule is the provision of two
special scales at the bottom of the groove, to which a bevelled
metal index or marker on the left end of the slide can be set.
The upper of these scales is for determining the efficiency of
dynamos and electric motors ; the lower for determining the loss
of potential in an electric circuit.
The Ferry Log.-log. Rule. — In the latest form of this rule the
log. -log. scales are arranged as indicated in Fig. 16, the E scale,
running from 1*1 to 10,000, being placed above the A scale of the
rule, and the — E or E' 1 scale running from 0'93 to 0*0001, below
the D scale of the rule. These scales are read in conjunction with
the B scales on the slide by the aid of the cursor.
The following tabular statement embodies all the instructions
required for using this form of log. -log. slide rule : —
When x is greater than 1.
x n Set 1 on B to x on E ; over n on B read x n on E.
x " Set 1 on B to x on E ; under n on B read xr n on E- 1
i^ i
x" Set n on B to x on E j over 1 on B read x" on E
x " Set n on B to x on E ; under 1 on B read x " on E _1
When x is less than 1.
x n Set 1* on B to x on E" 1 ; under n on B read x n on E" 1
x " Set 1 on B to x on E' ] ;' over n on B read x " on E
x H Set n on B to x on E- 1 ; under 1 on B read x n on E" 1
x • Set n on B to x on E -1 ; over 1 on B read x" on E
If 10 on B is used in place of 1 on B, read x 10 in place of
x n on E, and x 10 in place of x~" on E- 1 . If 100 on B is used, these
readings are to be taken as x m and x 10 ° respectively.
In rules with no — E scale the value of x~ n is obtained by the
usual rules for reciprocals. We may either determine a? and find
its reciprocal or, first find the reciprocal of x and raise it to the
n\X\ power. The first method should be followed when the
number x is found on the E scale.
Ex.— 3-45- 182 = 0-105.
Set 1 on C to 3*45 on E, and under 1*82 on C read 9*51 on C.
Then set 1 on B to 9*5 on A, and under index of A read 0-105 on B.
• H
92 THE SLIDE RULE:
When x is less than 1 the second method is more suitable.
Ex.— 0-23- 1 ' 77 = ($^\ 1>77 = 4 -35 1,77 = 13 '5
Set 1 on B to 0*23 on A, and under index of A read — — = 4*35 on B.
yj'Zo
Set 1 on C to 4 '35 on E, and under 1*77 on C read 13*5 on E.
As with the Davis rule, the exponent scale C will be read as
y^th its face value if its it.H. index (10) is used in place of 1.
SPECIAL TYPES OF SLIDE EULES.
In addition to the new forms of log. -log. slide rules pre-
viously described, several other arrangements have been recently
introduced, notably a series by Mr. A. Nestler, of Lahr
(London : A. Fastlinger, Snow Hill). These comprise the
"Rietz," the "Precision," the "Universal," and the "Fix"
slide rules.
The Rietz Rule. — In this rule the usual scales A, B, C, and D,
are provided, while at the upper edge is a scale, which, being three
times the range of the D scale, enables cubes and cube roots to be
directly evaluated and also r$ and n*.
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 20in. rule is obtainable in a length
of lOin. This is effected by dividing a 20in. (50cm.) 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 50cm. scales on
the slide, is virtually twice the length of that ordinarily obtainable
in a lOin. rule. The same remark applies to the trigonometrical
scales on the under face of the slide. Both the sine and tan unit
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 60cm. 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
A PRACTICAL MANUAL 93
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 Eule. — 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°?i. 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 Eule. — This is a standard rule in all respects,
except that the A scale is displaced by a distance - 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.
94 THE SLIDE RULE:
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 stoek. 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-Gravet 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 lO to 10 n + 1 , and the upper pair as running from ^10 x 10n to
^/lOxlO 71 " 1 " 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 axbxc 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 arc fixed to the
slide extending over the face of the rule as shown in the illustra-
tion. As each scale section is 30cm. in length, the upper lines
A PRACTICAL MANUAL
05
i=",;~??r
correspond to a single scale of nearly 4ft., and
the lower set to one of nearly 8ft. in length,
giving a correspondingly large increase in the
number of sub-divisions 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 5J 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 is used for
*"" units, that headed 4 for tens, and so on ; one
2 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.
The Multiplex Slide Rule diners 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.
96 ?HE SLIDE RULE:
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.
LONG -SCALE SLIDE RULES
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 20in. 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 along scale laid in spiral form
upon a disc, as in Fearnley's Universal Calculator and Schuemian'a
Calculating Instrument ; and (3) the adoption of a long Bcale
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 ia
shown in Fig. 18, consists of a cylinder d capable of being moved
A PRACTICAL MANUAL
97
U] and down and around the cylindrical stock /, 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 41ft.
8in., 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
/*, 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.
In using the instrument a given number on d
is set to the fixed index 6, 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 fr, the fourth
term of the proportion will be found under a
or c. Of course, in multiplication, one factor
is brought to 6, 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 involv-
ing continuous multiplication, or combined multi-
plication 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 axbxc: — 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.
- a j
Fig. 18.
08
THE SLIDE RULE:
For division, as
. bring atoF; A or B to m ; 100 to A
4r: I
A or B to a ; 100 to A and read the quotient at F.
The maximum number of figures in the quotient is the differ-
ence 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
tliis 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 Dy 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. E. H. Smith, the scale-line, which
is 50in. 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 fin. in diameter
and 9|in. 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.
. 19.
A PRACTICAL MANUAL 90
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
mantissoe 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.
Tiiacher's Calculating Instrument, shown in Fig. 20, consists
of a cylinder 4in. in diameter and 18in. long, which canbe given
both a rotary and a longitudinal movement within an open frame-
work composed of twenty triangular bars. These bars are con-
nected 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
Fig. 20.
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 correspond-
ing 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.
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.
100
THE SLIDE RULE.
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
Fig. 21.
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.
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 mult iplicatioo
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.
.4 PRACTICAL MANUAL
101
CIRCULAR CALCULATORS.
Although the lOin. 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 waist-
coat 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
Fig. 22.
Fig. 23.
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.
The Boucher Calculator. — This circular calculator resembles
a stem-winding watch, being about 2in. in diameter and tV n * m
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,
102 THfi SLIDE RULE:
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 ii.vd
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 4f in. 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
A PRACTICAL MANUAL 103
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-ofT 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 bring-
ing 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 j 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, repre-
senting 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 residt under the fixed index.
The foregoing method being an inversion of the rule for multi-
plication, 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 j = 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.
104 THE SLIDE RULE:
a x b x c
Combined Multiplication and Division, as = #, can be
readily performed, while cases of continued multiplication evidently
come under the same category, since axbx c=— ^ r— = x.
Ixi
, a , ..axlxlxl
feuch cases as — x are regarded as =x ; while
mxnxr mx n xr
—x is similarly modified, taking the form — =x. In
nn * ° m x 1
all cases the expression must be arranged so that there is one
more factor in the numerator than in the denominator, l's being
introduced as often as required. The simple operations of multi-
plication and division involve a similar disposition of factors, since
from the rules given it is evident that m x n is actually regarded as
TYi x n m m x 1
— - — , while — becomes in effect . It is important to note the
1 n n r
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 —x are taken in the order: — 1st factor of
r m x n
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.
39x14-2x6-3 , ,
Ex - 1-37x19 = 134 '
Commence by setting 39 to the index, and the pointer to 1*37 ;
bring 14*2 to the pointer ; pointer to 19 ; 6'3 to the pointer, and
read the result 134 at the index.
It should be noted that after the first factor is set to the fixed
index, the 'pointer is set to each of the dividing factors as tluy
enter into the calculation, while the dial is moved for each of the
multiplying factors. Thus the dial is first moved (setting the
first factor to the index), then the pointer, then the dial, and
so on.
Number of Digits in the Residt. — If rules are preferred to the plan
of roughly estimating the result, the general rules given on pages
21 and 25 should be employed for simple cases of multiplication
and division. For combined multiplication and division, modify
A PRACTICAL MANUAL 105
the expression, if necessary, by introducing 1's, as already explained,
and subtract the sum of the denominator digits from the sum
of numerator digits. Then proceed by the author's rule, as
follows : —
Always turn dial to the left ; i.e., against the hands of a watch.
Note dial movements only ; ignore those of the pointer.
Each time 1 on dial agrees with or passes fixed index, add 1 to the
above difference of digits.
Each time 1 on dial agrees with or passes pointer, deduct 1 from
the above difference of digits.
Treat continued multiplication in the same way, counting the
l's used as denominator digits as one less than the number of
multiplied factors.
8'6 x 0-73x1-02 » Mr „- mM -,
Ex - 3-5,x0-23 =7 ' 95 P«*»+>
Set 8'6 to index and pointer to 3*5. Bring 0'73 to pointer
(noting that 1 on the scale passes the index) and set pointer to
023. Set 1'02 to pointer (noting that 1 on the scale passes the
pointer) and read under index 7 "95. There are 1 + + 1 = 2
numerator digits and 1 + = 1 denominator digit ; while 1 is to be
added and 1 deducted as per rule. But as the latter cancel, the
digits in the result will be 2-1=1.
When moving the dial to the left will cause 1 on the dial to
pass both index and pointer (thus cancelling), the dial may be
turned back to make the setting.
It will be understood that when 1 is the first numerator, and 1
on the dial is therefore set to the index, no digit addition will be
made for this, as the actual operation of calculating has not been
commenced.
In the Stanley-Boucher calculator (Fig. 23) a small centre scale
is added, on which a finger indicates automatically the number of
digits to be added or deducted ; the method of calculating,
however, differs from the foregoing. To avoid turning back
to at the commencement of each calculation, a circle is
ground on the glass face, so that a pencil mark can be made
thereon to show the position of the finger when commencing a
calculation.
To Find the Square of a Number.— Set the number, on one or
other of the square root scales, to the index, and read the required
square on the ordinary scale.
106 THE SLIDE RULE:
To Find the Square Root of a Number.— Set the number to the
index, and if there is an odd number of digits in the number, read
the root on the inner circle ; if an even number, on the second
circle.
To Find the Cube of a Number. — Set 1 on the ordinary scale to
the index, and the pointer (on the back dial) to the number on
one of the three cube-root scales. Then under the pointer read
the cube on the ordinary scale.
To Find the Cube Root of a Number. — Set 1 to index, and pointer
to number. Then read the cube root under the pointer on one of
the three inner circles on the back dial. If the number has
1, 4, 7, 10 or -2, -5, etc., digits, use the inner circle.
2, 5, 8, 11 or- 1, -4, etc., „ „ second circle.
3, 6, 9, 12 or -0,-3, etc., ,, „ third circle.
For Powers or Roots of Higher Denomination. — Set 1 to index,
the pointer to the number on the ordinary scale, and read on the
outer circle on the back dial the mantissa of the logarithm. Add
the characteristic (see p. 46), multiply by the power or divide
by the root, and set the pointer to the mantissa of the result on
this outer circle. Under the pointer on the ordinary scale read
the number, obtaining the number of figures from the character-
istic.
To Find the Sines of Angles. — Set 1 to index, pointer to the
angle on the outer circle, and read under the pointer the natural
sine on the ordinary scale ; also under the pointer on the outer
circle of the back dial read the logarithmic sine.
The Halden Calculex. — After the introduction of the Boucher
calculator in 1876, circular instruments, such as the Charpentier
calculator, were introduced, in which a disc turned within a fixed
ring, so that scales on the faces of both could be set together and
ratios established as on the slide rule. Cultriss's Calculating Disc
is another instrument on the same principle. The Halden Calculex,
of which half-size illustrations are given in Figs. 24 and 25,
represents a considerable improvement upon these early instru-
ments. It consists of an outer metal ring carrying a fixed scale
ring, within which is a dial. On each side of this dial are Hat
milled heads, so that by holding these between the thumb and
forefinger the dial can be set quickly and conveniently. The
protecting glass discs, which are not fixed in the metal ring but
A PRACTICAL MANUAL
107
are arranged to turn therein, carry fine cursor lines, and as these
an- <>n the side next to the scales a very close setting can be made
quite free from the effects of parallax. This construction not only
avoids the use of mechanism, with its risk of derangement, but
reduces the bulk of the instrument very considerably, the thickness
1 icing about ^in.
On the front face, Fig. 24, the fixed ring carries an outer evenly-
divided scale, giving logarithms, and an ordinary scale, 1 — 10, which
works in conjunction with a similar scale on the edge of the dial.
The two inner circles give the square roots of values on the main
scales as in the Boucher calculator. On the back face, Fig. 25, the
Fig. 24.
Fig. 25.
ring bears an outer scale, giving sines of angles from 6° to 90° and
an ordinary scale, 1 — 10, as on the front face. The scales on the
dial are all reversed in direction (running from right to left), the
outer one consisting of an ordinary (but inverse) scale, 1 — 10, while
the three inner circles give the cube roots of values on this inverse
scale. As the fine cursor lines extend over all the scales, a variety
of calculations can be effected very readily and accurately.
Sperry's Pocket Calculator, made by the Keuffel and Esser
Company, New York (Fig. 26), has two rotating dials, each with
its own pointer and fixed index. The S dial has an outer scale of
equal parts, an ordinary logarithmic scale, and a square-root
scale. The L dial has a single logarithmic scale arranged spirally,
in three sections, giving a scale length of 12|in. The pointers are
turned by the stem on pressing either of the small buttons shown,
but the gearing is such that both the L dial and its pointer rotate
108
THE SLIDE RULE:
three times as fast as the S dial and pointer. All the usual
calculations can be made with the spiral scale, as with the Boucher
calculator, and the result read oft' on one or other of the three
scale-sections. Frequently the point at which to read the result is
obvious, but otherwise a reference to the single scale on the
S dial will show on which of the three spirals the result is to be
found.
The Bother Calculator is an instrument which is of special
service for surveying calculations. It consists of a substantial
base board, upon which is mounted a rotatable disc of cardboard,
200 mm. in diameter. A single reference line on the base answers
S. Dial.
Fig. 26.
L. Dial.
the purpose of the fixed index of the Boucher calculator, but
by placing other gauge points on the base, the instrument can
be adapted to meet various special requirements. A radial
cursor of thin transparent celluloid extends well over the base
board.
The outer scale on the disc is a simple logarithmic scale, 1 — 10,
equivalent to a straight rule, 25in. in length. Next to this la a
scries of four short scales, placed end to end, enabling sines and
cosines to be obtained from tangents, secants and tangents from
sines and other similar trigonometrical transformations, t<> be
readily effected. Next in order is a complete scale giving tangents
and cotangents, while the innermost scale is evenly divided giving
logarithms. This calculator is supplied in four sizes by E. Biow,
Wurzburg.
A PRACTICAL MANUAL
109
SLIDE RULES FOR SPECIAL CALCULATIONS.
Engine Power Computer. — A typical example of special slide
rules is shown in Fig. 27, which represents, on a scale of about
half full size, the author's Power Computer for Steam, Gas, and
Oil Engines. This, as will be seen, consists of a stock, on the
lower portion of which is a scale of cylinder diameters, while the
upper portion carries a scale of horse-powers. In the groove
between these scales are two slides, also carrying scales, and cap-
able of sliding in edge contact w^ith the stock and with each other.
This instrument gives directly the brake horse-power of any
steam, gas, or oil engine ; the indicated horse-power, the dimensions
of an engine to develop a given power, and the mechanical efficiency
of an engine. The calculation of piston speed, velocity ratios of
Fig. 27.
pulleys and gear wheels, the circumferential speed of pulleys, and
the velocity of belts and ropes driven thereby, are among the
other principal purposes for which the computer may be employed.
The Smith-Davis Piecework Balance Calculator has two
scales, 11 feet long, having a range from Id. to £20, and marked
so that they can be used either for money or time calculations.
The scales are placed on the rims of two similar wheels and so
arranged that the divided edges come together. The wheels are
mounted on a spindle carried at each end in the bearings of a
supporting stand. The wheels are pressed together by a spring,
and move as one.
To set the scales one to the other, a treadle gear is arranged to
take the pressure of the spring so that when the fixed wheel is
held by the left hand the free wheel can be rotated by the right
hand in either direction. When the amount of the balance has
been set to the combined weekly w T age the treadle is released
110 THE SLIDE RULE:
locking the two wheels together, when the whole can be turned
and the amounts respectively due to each man read off opposite
his weekly wage. The Smith-Davis Premium Calculator is on the
.same principle but the scales are about 4 feet 6 inches long and
the wheels spring-controlled. Both instruments are supplied by
Messrs. John Davis & Son, Ltd., Derby.
The Baines Slide Eule. — In this rule, invented by Mr.
H. M. Baines, Lahore, four slides carrying scales are arranged to
move, each in edge contact with the next. The slides are kept in
contact and given the desired relative movement one to the other,
by being attached (at the back), to a jointed parallelogram. On
this principle which is of general application, the inventor has
made a rule for the solution of problems covered by Flam ant's
formula for the flow of water in cast-iron pipes: — V = 76*28<i 7 s f ,
in which s is the sine of the inclination or loss of head ; d the
diameter of the pipe in inches and V the velocity in feet per
second. The formula Q = AV is also included in the .scope of the
rule, Q being the discharge in cubic feet per second and A the
cross sectional area of the pipe in square inches.
Maitland's Hydraulic Slide Eule. — This rule, made by
Messrs. W. F. Stanley & Co., Ltd., has a single slide and is
designed to solve Kutter's formula for the flow of water in channels.
In this case the value of n, the co-efficient of roughness, has been
taken atO'013, and the discharges and velocities pertain to circular
sewers flowing half -full and to oval sewers (new form) flowing two-
thirds full.
Among other special slide rules, mention may be made of the
Princeps Slide Rule for artillery and field purposes, which enables
various problems connected with sighting, deflection, etc., to be
solved very readily ; and Best's Simplified Slide Ride in which the
upper scales are duodecimally divided, and the lower, decimally,
as usual. Duodecimal slide rules, of which many forms have been
introduced from time to time, enable twelfths (as inches or pence)
to be dealt with directly without converting them into equivalent
decimal values.
CONSTEUCTIONAL IMPEOVEMENTS IN SLIDE EULES.
The attention of instrument makers is now being given to the
devising of means for ensuring the smooth and even working of
the slide in the stock of the rule. In some cases very good results
A PRACTICAL MANUAL
111
are obtained by slitting the back of the stock to give more
elasticity. Mr. A. W. Faber provides a side spring which presses
against one edge of the slide, and ensures smooth working through-
out the whole length of the rule.
In the rules supplied by Messrs. John Davis & Son, and
shown in section in Fig. 28, a steel strip, slightly curved in cross-
section as shown at A, runs for nearly the full length of the
stock, being fastened to the latter at intervals. A fine groove
along the centre of the stock gives elasticity to the latter and
allows the spring plate to close the sides of the stock on to the
slide, giving a smooth and easily adjustable fit. In Fig. 29 is
shown the same makers' special slide rule for hot or damp
climates. The back is made as above described, and in addition
Fig. 29.
Fig. 30.
three adjusting screws are provided, applied as shown. By
regulating these the slide may be made to travel smoothly from
end to end.
In the rule made by the Keuffel and Esser Company of New
York, one strip is made adjustable, allowing the fit to the slide to
be regulated as desired (Fig. 30).
Nestler's rules are now provided with rubber insertions let in
the sides of the stock of the rule ensuring a smooth movement of
the slide. This is of especial advantage in the longer rules.
THE ACCUEACY OF SLIDE EULE EESULTS.
The degree of accuracy obtainable with the slide rule depends
primarily upon the length of the scale employed, but the accuracy
of the graduations, the eyesight of the operator, and, in particular,
his ability to estimate interpolated values, are all factors which
112 THE SLIDE RULE.
affect the result. Using the lower scales and working carefully
the error should not greatly exceed 0*15 per cent, with short calcu-
lations. With successive settings, the discrepancy need not
necessarily be greater, as the errors may be neutralised ; but with
rapid working the percentage error may be doubled. However,
much depends upon the graduation of the scales. Rules in which
one or more of the indices have been thickened to conceal some slight
inaccuracy should be avoided. The line on the cursor should be
sharp and fine and both slide and cursor should move smoothly
or good work cannot be done. Occasionally a little vaseline or clean
tallow should be applied to the edges of the slide and cursor.
That the percentage error is constant throughout the scale is
seen by setting 1 on C to 1*01 on D, when under 2 is 2 '02 ; under
3, 3*03 ; under 5, 5*05, etc., the several readings showing a uniform
error of 1 per cent.
A method of obtaining a closer reading of a first setting or of a
result on D has been suggested to the author by Mr. M. Ainslie,
B.Sc. If any graduation, as 4 on C, is set to 3 on D, it is seen
that 4 main divisions on C (40-44) are equal in scale length to 3
main divisions on D (30-33). Hence, very approximately, 1 division
on C is equal to 0*75 of a division on 1), this ratio being shown, of
course, on D under 10 on C. Suppose v/4'3 to be required. Set-
ting the cursor to 4*3 on A, it is seen that the root is something
more than 2*06. Move the slide until a main division is found on
C, which exactly corresponds to the interval between 2 and the
cursor line, on D. The division 27-28 just fits, giving a reading
under 10 on C, of 74. Hence the root is read as 2*074. For the
higher parts of the scale, the subdivisions, 1-1*1, etc., are used in
place of main divisions. The method is probably more interesting
than useful, since in most operations the inaccuracies introduced
in making settings will impose a limit on the reliable figures of the
result.
For the majority of engineering calculations, the slide rule will
give an accuracy consistent with the accuracy of the data usually
available. For some purposes, however, logarithmic section j>
(the use of which the author has advocated for the last twenty
years) will be found especially useful, more particularly in calcula-
tions involving exponential form aloe.
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ADVERTISEMENTS
Second Edition. Crown 8vo, Is. Id. post free
LOGARITHMS FOR BEGINNERS
FOR a full and intelligent appreciation of the Slide Rule and its
various applications an elementary knowledge of logarithms is
necessary. All that, is required will be found in this little work, which
gives a simple, detailed, and practical explanation of logarithms and
their uses, particular care having been taken to elucidate all difficult
points by the aid of a number of worked examples.
Third Edition. Crown 8vo, 3s. net
THE INDICATOR:
ITS CONSTRUCTION AND APPLICATION
In this work all the various modern types of indicator are fully described
and illustrated, while the attachment and actuation of the instrument is
explained in far greater detail than in any other work. Much useful
information is given on the adjustment and manipulation of the instrument.
Third Edition. Crown 8vo, 3s. net
THE INDICATOR DIAGRAM:
ITS ANALYSIS AND CALCULATION
In this work the analysis of the indicator diagram is undertaken in a thor-
oughly systematic and exhaustive manner. Diagrams from compound
engines, from gas and oil engines, and from air compressors, pumps, etc.,
are fully discussed and the best methods of diagram calculation described.
POWER COMPUTER
FOR
STEAM, GAS AND OIL ENGINES
Gives instantly and without calculation of any kind whatever:
The Indicated Horse-power of Steam, Gas, and Oil Engines.
The Brake Horse-power of Steam, Gas, and Oil Engines.
The Size of Engine necessary to develop any given power.
The Mechanical Efficiency of an Engine.
The Piston Speed of an Engine.
The Circumferential Speed of Wheels, or of Ropes, Belts,
&c., driven thereby.
Speed Ratios of Pulleys, Gearing, &c, &c.
Pocket Size, in neat case, with instructions, 5/- net ;
or in wood, with permanent Ivorine Scales, 10/6 net, post free
C. N. PICKWORTH, Fallowfield, MANCHESTER
ADVERTISEMENTS
W. H. HARLING
/Iftatbemattcal, drawing an& Surveying
instrument /Manufacturer
47 & 49 FINSBURY PAVEMENT
LONDON, E.C.
| &jot £■ - — •
HARLING's No. 2003.
"STANDARD"
10", 10/6; 15", 27/-; 2€
HARLING'S No. 2002.
"ELECTRICAL'
11", 13/6.
8
.--■ ij-
1 \.-"
pfe
6
1
: - I
1 1 .
11 .mi •.
s P
f ■ * y {*{ ®
L_ %
f 1 • J 9
f—
■— 1
"
11
*
1
mi
. . ■ V
Harlin 2004.
"CUBING"
10", 15/-; l.V, 32/-; 20", 60/-.
HAR LING ? S No. 2005.
"POCKET"
5" thin, 6/6.
Harlin . ' N . 2006.
"POCKET"
With Magnifier, 5", 8/6.
STUDENT'S SLIDE RULE, made of cardboard, with metal cursor
5", 1/1 ; 10", 2/1 (post free).
Other Slide Rules kept in stock—
"DAVIS, "FABERS, "FULLER, "LONG SCALE, "FERRY. "PRECISION
UNIVERSAL"
Also Watch Form—
"BOUCHER SYSTEM " "CALCULEX, ' "M. E. CALCULATOR
AUVKKTISKMENTiS
FOUNDED 1770
SLIDE RULE MANUFACTORY ESTABLISHED 1820
T 3 AYERNIER=GRAYET
(PAUL MICHON, Successor)
19 RUE MAYET, PARIS YI.
Awarded Gold Medals— Paris, 1878, 1889, 1900 ; Moscow. 1891 :
Brussels, 1897 ; Liege, 1905 ; Milan, 1900.
Slide Rules in 40 different forms
THE BEGHIN SLIDE RULE
(Awarded a (Jold Medal at the Tourcoing Exhibition, 1906)
Thin instrument enables many calculations to be made with one setting ;
traversing the slide to change indices is avoided, and more
accurate results are therefore obtained.
PRICES AND PARTICULARS ON APPLICATION
TAVERNIER-GRAVET SLIDE RULES
are obtainable through the leading English Instrument Dealers
The PHOSPHOR BRONZE CO., Ltd.
87 SUMMER ST., SOUTHWARK, LONDON, S.E.
PHOSPHOR BRONZE
(The original Cog Wheel and
Vulcan Brands).
Fot Slide Valves, Bearings,
Bushes, Pinions, Worms and
Worm Wheels, Motor Gear-
ing, etc.
BABBITT'S METAL
(Vulcan Brand).
NINE GRADES
PLASTIC METAL
(Vulcan Brand), for Lining
1 Ictrings, etc.
ANTI-FRICTION METALS
("White Ant" Brand).
Cheaper than Babbitt's.
PHOSPHOR COPPER and
PHOSPHOR TIN
(Cog Wheel Brand). The finest
quality made.
Rolled and Drawn Phosphor and other Bronze
Castings in Phosphor Bronze, Gun Metal and Aluminium, to patterns oi draw-
ings J machined if required.
Chill Bar and Tube Castings for turning into small Bearings and Bushes,
Spindles and Small Pump Rods.
ADVERTISEMENTS
Tavernier-Gravet Slide Rule
CELLULOID SCALES . . . GLASS CURSOR
£ s.
10 in. (2Sc/m) Tavernier-Gravdt Slide Rule, engine divided scales
on white celluloid, glass cursor in metal mount .0106
Case extra, Is. 6d. ; postage 3d., or to the Colonies, Is. 6d.
5 in. (12*5 c/m) Tavernier-Gravet Slide Rule, celluloid scale,
glass cursor . . . . . . . . 10 6
Do. do , do. 15
Case extra, 2s.
Do. do., do.
in wood case . . . . . . . 2 10
14 in. (36 c/m)
20 in. (50 c/m)
SOLE MAKER OF
HALL'S NAUTICAL SLIDE RULE
Designed to save the navigator the labour of arithmetical calculation and
the use of tables when reducing the ex-meridian and finding corresponding
errors in Latitude and Longitude, etc., etc.
With instructions in case . . . 17s. 6d.
SOLE MAKER OF
THE R.H.S. CALCULATOR
A spiral logarithmic scale, 50 inches long, wound upon a cylinder, 8 inches
long and about 0*75 inches in diameter. A fourth figure in a number can be
read by eye division. Designed and patented by Professor Robert H. Smith.
(New Model in preparation)
CATALOCUE. PART IV. Surveying, Mathematical and Nautical Instruments, including
Theodolites, Tacheometers, Levels, Pocket Telemeters, Clinometers,
Prismatic and I, uminous Compasses. POST FREE ON APPLICATION •
J. H. STEWARD,
OPTICIAN to the BRITISH and
FOREIGN GOVERNMENTS
406 & 457 Strand, LONDON
Telegraphic Address: "TELEMETER,'' LONDON
TELEPHONE: 1867 CERRARD
ADVERTISEMENTS
EUGENE DIETZGEN GO.
NEW YORK CHICAGO
214=220 East 23rd Street 181 Monroe Street
NEW ORLEANS SAN FRANCISCO
615 Common Street 16=28 First Street
TORONTO, CAN.
10 Shuter Street
Manufacturers and Importers
DRAWING MATERIALS
SURVEYING INSTRUMENTS
SLIDE RULES, Etc.
All the Latest Improved Styles of SLIDE RULES
particularly
THE MULTIPLEX . MACK
PRECISION . . RIETZ
UNIVERSAL . . FULLER
THE HALDEN CALCULEX
SEXTON'S OMNIMETERS
Complete 475 Page CATALOGUE sent on request to
DEALERS or PROFESSIONALS of Good Standing.
ADVKKTI8KMENTS
- ? -^T
ANDERSON'S
IMPROVED
SLIDE RULE
GREAT INCREASE IN ACCURACY
T^ Y this invention Lieut.-Col. Anderson has supplied the
*-' long-felt want of a SLIDE RULE sufficiently accurate
for all practical purposes without being unduly cumber-
some. This improved Slide Rule is eight times as accurate
as an ordinary Slide Rule of the same length, and equally
simple to use. To Engineers, Electricians, Architects, Sur-
veyors, Accountants, Builders, Excise Officials, Statisticians,
and others who have to deal with laborious calculations, the
improved instrument is indispensable. A special handbook
with useful conversion and other tables accompanies each Slide
Rule, which is packed in a neat cardboard box, all complete.
Price 21s.
Box and Postage, 6d. extra -
Full particulars sent on application
C. F. CASELLA & CO.
Scientific Instrument Makers
II, 13 and 15 ROCHESTER ROW, VICTORIA STREET
LONDON S.W.
ADVERTISEMENTS
A. W. FABER
149 QUEEN VICTORIA STREET
LONDON, E.G. - -
Manufactories in Germany, France and United States.
Houses in London, Paris, Berlin and Newark, U.S.A.
Gold and First-class Prize Medals.
GRAND PRIX (Highest Award) PARIS, 1900
GRAND PRIZE (Highest Award) St LOUIS, 1904.
SUPERIOR LEAD AND COLOURED PENCILS
DRAWING MATERIALS FOR ENGINEERS, ARCHITECTS
AND SURVEYORS.
A. W. FABER'S
IMPROVED CALCULATING RULES.
"The utmost accuracy, combined with the most perfect workmanship
and finish."
DESCRIPTIVE PAMPHLET ON APPLICATION.
A. W. FABER'S productions may be obtained from all
Dealers in Drawing Instruments or Stationers.
PENCIL MANUFACTORY ESTABLISHED 1761
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STAMPED BELOW
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THIS BOOK ON THE DATE DUE. THE PENALTY
WILL INCREASE TO 50 CENTS ON THE FOURTH S
DAY AND TO $1.00 ON THE SEVENTH DAY
OVERDUE.
USE
[AT
]._ -£3*4,
' ' /
SPECIAL
Stren
Pliabf
Light-
Resis
THE MOJ
Uffl*^
use
flQV V
"NOV 1 7 1958 L U
Hart's
fcV
*±
F -^
THC , 1
Telephon yib^
Rgc»r>
J
0CT31-6?
- ^^iV QtlftT
BEC'OtD $JP1?71-2PW>2 4
jr -jjt
_ft3l
TT
LD 21-100ro-7,'40 (6936s)
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cr\ Extra -Strong Paper, w)h a • plain, margin to eacfv
sneer, and maaeup info-pads • \% x 9.
^JuJI particulars jj specimen -snas/s sent-on- applicatioro
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