How to Use Trig Slide Rules
5
" be _— -
/
i. a e
;
\
if
+
a
rf
a
p
i \
;
4.
x
>
aod
CHICAGO 5, ILLINOIS « ALHAMBRA, CALIFORNIA
:
by MAURICE L. HARTUNG
) _ Professor of the
: Teaching of Mathematics
THE UNIVERSITY OF CHICAGO
FORM M-12 R pear jubilee
PRINTED IN U.S.A. ‘ i /
PREFACE
A computer who must make many difficult calculations usually has a
slide rule close at hand. In many cases the slide rule is a very convenient
tool because by means of a few simple adjustments the calculations can
be carried through and the result obtained. One has only to learn to read
the scales, how to move the slide and indicator, and how to set them
accurately, in order to be able to perform long and otherwise difficult
calculations.
When people have difficulty in learning to use a slide rule, usually it is
not because the instrument is difficult to use. The reason is likely to be
that they do not understand the mathematics on which the instrument is
based, or the formulas they are trying to evaluate. Some slide rule
manuals contain relatively extensive explanations of the theory under-
lying the operations. In this manual such theory has deliberately been
kept to a minimum. It is assumed that the theory of exponents, of
logarithms, of trigonometry, and of the slide rule is known to the reader,
or will be recalled or studied by reference to formal textbooks on these
subjects. This is a brief manual on operational technique and is not
intended to be a textbook or workbook.
Some of the special scales described in this manual may not be avail-
able on your slide rule. All of the illustrations and problems shown can
be worked on the slide rule you purchased. However, the special scales
simplify the calculations. Pickett & Eckel Slide Rules are available with
all of the special scales shown in this manual and also with Log Log fea-
tures not described here.
\
COPYRIGHT 1960 - PICKETT & ECKEL INC. - CHICAGO 5, ILL.
TABLE OF CONTENTS
Part 1—SLIDE RULE OPERATION PAGE
Introduction. 2.0... cece eee eee etna nen erecees 3
How to Read the Scales... 1.0... ccc cece eee eee en eees 4
Multiplication. ....... 06.6 cece nec ee tener eres er eenens 7
Decimal Point Location. ... 0... 0c. cece ence eee e eect ennee 9
DIVISION. 6:06 cose bois ciate cere en eeeeeele eee ne O88 eee ane omen alae 10
Continued Products.........00 ccc ccc ce eece se ncnerecernes 13
Combined Multiplication and Division...............+++-- 14
Proportion... 0.0... ccc cece eee e eee nee n ener eeeseerens 15
Part 2—USE OF CERTAIN SPECIAL SCALES
The Cl and DI Scales... 10.0.0... cece eee eee eet n renee 19
The CF and DF Scales. ...... 2... ccc ccc cece ence tere enees 22
The CIF Scale... . icc ccc cc cece te eee n reece ener 23
The A and 8 Scales: Square Roots and Squares.......... 24
The K Scale: Cube Roots & Cubes.............- suererereve amano 27
Combined Operations: Square Roots and Squares....... 29
Combined Operations: Cube Roots and Cubes........... 31
The L Seale: Logarithms............2c0 cece ence ren eeeees 32
The §, T, and ST Scale: Trigonometry...............2+005 34
Special Graduations.............. ccc ce cece eee eeeneeenes 40
Combined Operations... ...........00ee cece eee ene en enes 42
Part 3—APPLICATIONS OF TRIGONOMETRY
Using the Law of Sines. . 2.2.0... cece cece ee eee e enone 43
Complex Numbers and Vectors. .........:ee eee eeeeeeee 45
Solving Triangles When Two Sides and the
Included Angle are Known...........-eees cece eeeenees 51
Solving Triangles When Three Sides are Known......... 53
Part 4—THE PRINCIPLES UNDERLYING SLIDE RULES
Scales for Addition. ............. cece sneer eee ee en enenee 54
Scales for Multiplication. ........... 0c cere eee ener eens 55
How Short Scales Can be Used.......... 0.0 ccc cere cree 57
How Logarithmic Scales are Made..............-5seeeeee 58
The Principle of the Slide Rule Explained in
Terms of Logarithms. ............. 2c cece cence eee ee 59
: PART I. SLIDE RULE OPERATION
INTRODUCTION
The slide rule is a simple tool for finding numerical answers to involved
mathematical problems. To solve problems easily and with confidence
it is necessary to have a clear understanding of the operation of your
slide rule. Speed and accuracy will soon reward the user who makes a
careful study of the scale arrangements and the manual.
The table below shows some of the mathematical operations which can
be done easily and quickly with an ordinary slide rule.
htnbtatctnbetaed
OPERATIONS INVERSE OPERATIONS
Multiplying two or more numbers | Dividing one number by another
Squaring a number pe inding the square root of a number
Cubing a number iP inding the cube root of a number
Finding the logarithm of a number! Finding a number whose logarithm
is known
Finding the sine, cosine, or tangent Finding an angle whose sine, cosine,
i of an angle or tangent is known
Various combinations of these operations (such as multiplying two
i numbers and then finding the square root of the result) are also easily
done.
The slide rule consists of three parts: (1) the body (upper and lower
fixed bars); (2) the slide; (3) the cursor or indicator. The scales on the
body and slide are arranged to work together in solving problems. The
hairline on the indicator is used to help the eyes in reading the scales and
Maden
i in adjusting the slide.
f
: Fig. 1 Q O14)
i BODY
SLIDE HAIRLINE
BODY
| CURSOR, OR INDICATOR I ©
F Each scale is named by a letter (A, B, C, D, L, 8, T) or other symbol at
the end.
In order to use a slide rule, a computer must know: (1) how to read the
seales; (2) how to “set” the slide and indicator for each operation to be
= MODEL 1030 TRIG MODEL 902 SIMPLEX TRIG done; ‘and (3) how to determine the decimal point in the result.
[2] ; [3]
HOW TO READ THE SCALES
The scale labeled C (on the slide) and the scale D (on the body) are
used most frequently. These two scales are exactly alike. The total length
of these scales has been separated into many smaller parts by fine lines
called “graduations.”
Some of these lines on the D scale have large numerals (1, 2, 3, etc.)
printed just below them. These lines are called primary graduations. On
the C seale the numerals are printed above the corresponding gradua-
tions. A line labeled 1 at the left end is called the left index. A line
labeled 1 at the right end is called the right index.
Fig. 2
p- {#11 conga ug mvo 141] Y Le ge TU 00) gee
PRIMARY GRADUATIONS
LEFT INDEX | RIGHT INDEX
Next notice that the distance between 1 and 2 on the D scale has been
separated into 10 parts by shorter graduation lines. These are the second-
ary graduations. (On 10 inch slide rules these lines are labeled with
smaller numerals 1, 2, 3, ete. On 6 inch rules these lines are not labeled.)
Each of the spaces between the larger numerals 2 and 3, between 3 and 4,
and between the other primary graduations is also sub-divided into 10
parts. Numerals are not printed beside these smaller secondary gradua-
tions because it would crowd the numerals too much.
Fig. 3
b MCU IRL UL CA COCO PO RR MY A
LEFT INDEX 17 44 RIGHT INDEX
SECONDARY GRADUATIONS
When a number is to be located on the D scale, the first digit is located
by use of the primary graduations. The second digit is located by use of
the secondary graduations. Thus when the number J7 is located, the 1
at the left index represents the 1 in 17. The 7th secondary graduation
represents the 7. When 44 is to be located, look first for primary gradua-
tion 4, and then for secondary graduation 4 in the space immediately
to the right.
There are further sub-divisions, or tertiary graduations, on all slide
rules. The meaning of these graduations is slightly different at different
parts of the scale. It is also different on a 6 inch slide rule than on a 10
inch rule. For this reason a separate explanation must be given for each.
[4]
Tertiary graduations on 10 inch rules.
The space between each secondary graduation at the Jeft end of the
rule (over to primary graduation 2) is separated into ten parts, but these
shortest graduation marks are not numbered. In the middle part. of the
rule, between the primary graduations 2 and 4, the smaller spaces be-
tween the secondary graduations are separated into five parts. Finally,
the still smaller spaces between the secondary graduations at the right
of 4 are separated into only two parts.
To find 173 on the ID scale, look for primary division 1 (the left index).
Then look for secondary division 7 (numbered). Then look for smaller
subdivision 3, which is not numbered, but found as the 3rd very short
graduation to the right of the longer graduation for 7.
49 173
Fig. 4 246 247
; py ‘ aig ee te
LEFT INDEX
Similarly, 149 is found as the 9th small graduation mark to the right of
the 4th secondary graduation mark to the right of primary graduation 1.
To find 246, look for primary graduation 2, then for the 4th secondary
graduation after it. (the 4th long line), then for the 3rd small graduation
after it. The smallest spaces in this part of the scale are read as fifths.
Since 2 = /4,, then the third graduation, marking fhree fifths, is at the
same point as six tenths would be.
On the scale below are some sample readings for 10 inch scales.
Fig. 5
o . 8 A oF E
D [RCO ACEP RSL RR ALE A EAL
H G |
penne FORTS THUNK ;
A: 195 F: 206
B: 119 Gi: 465
C: 110 H: 402
]): 101 J: 694
E: 223 J: Y87
[a]
Tertiary graduations on 6 inch rules.
The space between cach secondary graduation at the left end of the
rule (over to primary graduation 2) is separated into five parts. In the
middle of the ruje, between the primary graduations 2 and 5, the
smaller spaces between the secondary graduations are separated into Lwo
parts. Finally, the still smaller spaces between the secondary graduations
at the right of 5 are not subdivided.
To find 170 on the D-scaie, look for the primary division 1 (the left
index), then for the 7th secondary graduation.
The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, used in writing numbers are
called digits. One way to describe a number is to tell how many digits are
used in writing it. Thus 5+ is a “two-digit number”, and 1,348,256 is a
“seven-digit number.” In many computations only the first two or three
digits of a number need to be used to get an approximate result which is
accurate enough for practical purposes. Usually not more than the first
three digits of a number can be “set” on a six inch slide rule scale, In
many practical problems this degree of accuracy is sufficient. When
greater accuracy is desired, a ten inch slide rule is generally used.
For an explanation of how these scales are constructed, see Page 58,
Fig. 6
© Veprificrstuentuitieuliniiin vandal EU te dele Letpdabete Mat tilitatl tf totale cr LBP catiaTinatalfiaidiahad : MULTIPLICATION
LEFT INDEX RIGHT INDEX Numbers that are to be multiplied are called factors. The result: is
146 a7 170 called the product. Thus, in the statement 6 X 7 = 42, the numbers 6
and 7 are factors, and 42 is the product.
® Exampie: Multiply 2 x 3.
Setting the Scales: Set the left index of the C scale on 2 of the D scale. Find
3 on the C scale, and below it read the product, 6 on the D scale.
Think: The length for 2 plus the length for 3 will be the length for the product.
This length, measured by the D scale, is 6.
To find 146, look for the primary graduation 1, then for the 4th sec-
ondary graduation after it (the 4th secondary line), then for the 3rd
small graduation after it. The smallest spaces in this part of the scale are
read as fifths. Since 3/5 = 6/10, then the third graduation, marking
three fifths, is at the same point as siz tenths would be.
The number 147 would be half of a small space beyond 146. With the
aid of the hairline on the runner the position of this number can be
located approximately by the eye. The smal! space is mentally “split?” in Fig. 8 ily:
half. . t LENGTH OF FACTOR 2 ON C SCALE {
The number 385 is found by locating primary graduation 3 and then LEFT 1H0e j _
secondary graduation 8 (the 8th long graduation after 3). Following this, OF veya Hi iia hina sa aa
one observes that between secondary graduations 8 and 9 there is one LNGTN OF FACTOR 2 ON D SCA | |
short mark. Think of this as the “5 tenths” mark, which represents 385. LENGTH OF 2 AUS LENGTH OF 3. IS LENGTH OF PRODUCT 6 ON D SCALE
The location of 383 can be found approximately by mentally “splitting”
the space between 380 and 385 into fifths, and estimating where the 3rd
“fifths” mark would be placed. It would be just a little to the right of
halfway between 380 and 385.
On the scale below are some sample readings for 6 inch scales.
Fig. 7
A: 104 F: 276
B: 128 G: 445
C: 143 H: 555
D: 179 I: 731
E: 205 J: 970
[6]
® Examp.e: Multiply 4 X 2.
Setting the Scales: Set the left index of the C scale on 4 of the D scale. Find
2 on the C scale, and below it read the product, 8, on the D scale.
Think: The length for 4 plus the length for 2 will be the length for the product.
This length, measured by the D scale, is 8. .
Rule for Multiplication: Over one of the factors on the D
scale, set the index of the C scale. Locate the other factor on the
C scale, and directly below it read the product on the D scale.
This rule may be shown in the form of a diagram as follows.*
; _ eis y
To find P=z-y, set: ala | P°
= : Co a 3
To find P=2X3, set: Do | P°
*This device for showing slide rule settings has been used for many years, See, for example, Lipka,
Joseph, Graphical and Mechanical Computation. John Wiley and Sons, Inc., (New York, 1918), p. 10.
[7]
@ Exampie: Multiply 2.34 X 36.8.
Estimate the result: First note that the result will be roughly the same as
2 X 40, or 80; that is, there will be two digits to the left of the decimal point.
Hence, we can ignore the decimal points for the present and multiply as
though the problem was 234 X 368.
Set the Scales: Set the left index of the C scale on 234 of the D seale. Find
368 on the C seale and read product 861 on the D seale.
Think: The length for 234 plus the length for 368 will be the length for the
product. This length is measured on the ID scale. Since we already knew the
result was somewhere near 80, the product must be 86.1, approximately.
® TixampLe: Multiply 28.3 X 5.46.
Note first that the result will be ahout the same as 30 X 5, or 150. Note
also that if the left index of the C scale is set over 283 on the D scale, and
546 is then found on the C scale, the slide projects so far to the right of the
body that the D scale is no longer below the 546. When this happens, the
other index of the C scale can be used. That is, set the right index on the C
scale over 283 on the D scale. Find 546 on the C scale and below it read
the product on the D seale. The product is 154.5.
2. Place hairline
over multiplicand (5,46)
on a
* if
Fig. 9
‘ I dal cretelattatantatatendbisaind
ctciottliranffila
PROBLEM SET I
1. 15 X37 2, 4.2 X 19
3. 62.1 K 3.9 4. 9.54 X 16.7
5. 1.85 X @ 6. 20.4 X 410
7. 9.3 X 8.7 8. 71.5 & 5.05
9. 53.6 XK 2.3 10. 3.76 X 18.2
M1. 298 X 0.26 12. 18.3 X 0.54
13. 0.81 X 92.2 14. 112 x 824
15. 0.29 K 0.31 16. 0.0215 X 3.79
DECIMAL POINT LOCATION
In the discussion which follows, it will occasionally be necessary to
refer to the number of “digits” and number of ‘zeros’”’ in some given
numbers.
When numbers are greater than 1 the number of digzts to the left of the
decimal point will be counted. Thus 734.05 will be said to have 3 digits.
Although as written the number indicates accuracy to five digits, only
three of these are at the left of the decimal point.
Numbers that are less than 1 may be written as decimal fractions.*
Thus .673, or six-hundred-seventy-three thousandths, is a decimal
fraction. Another example is .000465. In this number three zeros are
written to show where the decimal point. is loeated. One way to describe
Al ,
ATTIRE TA Le if
TUTTI UME Teg aT ET Hii ig i ee ¢
Lo Pitas Sriattiggs oat oe oiokd gages ont oe odegae url noes ws in lng, ST such a number is to tell how many zeros are written to the right of the
Henge rae et ee iat as $ decimal point before the first non-zero digit occurs.
1 mes 1: a ATT i : ' iuill} H .
EUUL, 1
yf nEIReEg eee va
3. Read result (154.5) 1, Set right index of C scale
on D scale. over multiplier (28.3) on D scale.
ETP TES [TT aye HT
In scientifie work a zero is often written to the left of the decimal point,
as in 0.00541. This shows that, the number in the units’ place is definitely
0, and that no digits have been carelessly omitted in writing or printing.
The zeros will neé be counted unless they are (a) at the right of the
| Haren bf re: nme
These examples illustrate how in simple problems the decimal point
can be placed by use of an estimate. Other methods of placing the
decimal point will be explained later in this manual.
In doing these simple examples it is not really necessary to use the
hairline on the cursor. However, most experienced slide rule users do use
it. Thus to multiply 2.34 X 36.8, they first place the hairline over 234
on D. Then they move the slide so that the index 1 of the C scale is under
the hairline. Then they move the cursor to place the hairline over 368 of
the C scale. The answer 86.1 is then read under the hairline on the D
scale.
To prevent the slide from changing position while the hairline is being
moved, pinch the upper and lower bars of the body together. They are
slightly flexible and will hold the slide until the pressure is released.
Later-sections of this mamual will explain how to do multiplication by
using other scales. For some examples the use of these other scales makes
the work easier. Use the C and D scales for Problem Set L. (Answers are
given in the back of this manual.)
[8]
a neg
decjma! point, (b) before or at the left of the first non-zero digit, and (c)
are not between other digits. The number 0.000408 will be said to have
3 zeros (that is, the number of zeros between the decimal point and the 4).
Method I. In many, perhaps a majority, of the problems met in
genuine applications of mathematics to practical affairs, the position of
the decimal point in the result can be determined by what is sometimes
called “common sense.” There is usually only one place for the decimal
point in which the answer is “reasonable” for the problem. Thus, if the
calculated speed in miles per hour of a powerful new airplane comes out
to be 4833, the decimal point clearly belongs between the 3’s, since 48
m.p.h. is too small, and 4833 m.p.h. is too large for such a plane. In some
cases, however, the data are such that the position of the point in the
final result is not easy to get by inspection.
Method I. Another commonly used method of locating the decimal
point is by estimation or approximation, For example, when the slide rule
is used to find 133.4 X 12.4, the scale reading for the result is 1655, and
*Only positive real numbers are being considered in this discussion.
[9]
the decimal point is to be determined. By rounding off the factors to
133.0 X 10.0, one obtains 1330 by mental arithmetic. The result would
be somewhat greater than this but certainly contains four digits on the
left of the decimal point. The answer, therefore, must be 1635.
Method ITI. In scientific work numbers are often expressed in slan-
dard form. For example, 428 can be written 4.28 X 10°, and 0.00395 can
be written as 3.95 & 107%. When a number is written in standard form
it always has two factors. The first factor has one digit (not a zero) on
the left of the decimal point, and usually other digits on the right of the
decimal point. The other factor is a power of 10 which places the decimal
point in its true position if the indicated multiplication is carried out.
In many types of problems this method of writing numbers simplifies the
calculation and the location of the decimal point.
When a number is written in standard form, the exponent of 10 may be
called “the characteristic.” It is the characteristic of the logarithm of the
number to base 10. The characteristic may be either a positive or a nega-
tive number. Although the rule below appears long, in actual practice
it may be used with great ease.
Rule. To express a number in standard form:
(a) place a decima! point at the right of the first non-zero
digit.*
(b) start at the right of the first non-zero digit in the original
number and count the digits and zeros passed over in reaching
the decimal point. The result of the count is the numerical
value of the characteristic, or exponent of 10. If the original
decimal point is toward the right, the characteristic is positive
(+). If the original decimal point is toward the left, the char-
acteristic is negative (—). Indicate that the result of (a) is to be
multiplied by 10 with the exponent thus determined in (b).
© EXAMPLES:
Number in
Number standard form. Characteristic
(a) 5,790,000 5.79 X 10° 6
(b) 0.000283 2.83 X 10-4 —4
(c) 44 4.4 xX 10! 1
(d) 0.623 6.23 xX 10-* —1
(e) 8.15 8.15 x 10° 0
(f) 461,328 4.61328 * 105 5
(g) 0.0000005371 5.371 X 107 —7
(h) 0.0306 3.06 X 107 ~2
{i} 80.07 8.007 x 101 1
If a number given in standard form is to be written in “ordinary”
form, the digits should be copied, and then starting at the right of the
first digit the number of places indicated by the exponent should be
counted, supplying zeros as necessary, and the point put down. If the
exponent is positive, the count is toward the right; if negative. the count
[10]
5
is toward the left. This converse application of the rule may be verified
by studying the examples given above.
Consider now the caleulation of 5,790,000 X 0.000283. From examples
(a) and (b) above, this can be written 5.79 X 10° XK 2.83 & 107', or by
changing order and combining the exponents of 10, as 5.79 X 2.83 X 10?.
Then since 5.79 is near 6, and 2.83 is near 3, the product of these two
factors is known to be near 18. The multiplication by use of the C and D
scales shows it to be about 16.39, or 1.639 & 101. Hence, 5.79 & 2.83 X
102 = 1.639 & 10! & 102 = 1.639 & 10? = 1639. If, however, one has
5,790,000 + 0.000283, the use of standard form yields
5.79X1] OF = 6—-(—4) — 10
283x108 2 Ot X10 =2.04X 10.
In scientifie work the result would be left in this form, but for popular
consumption it would be written as 20,400,000,000. The general rule is
as follows.
Rule. To determine the decimal point, first express the num-
bers in standard form. Carry out the indicated operations of
multiplication or division, using the laws of exponents to
combine the exponents until a single power of 10 is indicated.
If desired, write out the resulting number, using the final
exponent of 10 to determine how far, and in what direction, the
decimal point in the coefficient should be moved.
The theory of exponents and the rules of operation with signed num-
bers are both involved in a complete treatment of this topic. In this
manual it is assumed that the reader is familiar with this theory. The
“standard form” method of placing the decimal point can be used with
all types of calculations. It is the most general, scientific, and reliable
method, but for many problems it requires skill in the use of algebra.
Method IV. Special scales, called Decimal-Keeper Seales, have been
devised to help in locating the decimal point. These scales are found
only on Decimal-Keeper slide rules. The use of these scales is explained
on pages 55 to 56 of this manual.
There are some other methods, but they are difficult to use and re-
member. This is especially true whenever the problem is more compli-
cated than a simple multiplication or division of two numbers. For these
reasons these methods will not. be explained here.
@ EXAMPLES:
(a) 34,900 X 0.0012. Use Method II, and think of this as about
’ 12
30,000 X 10,000
which is 36. Use the C and D scales to multiply 349 by 12, obtain-
ing 419. Since the answer must be near 36, it must be 41.9.
(b) 34,900 < 0.0012. If Method III is used, write the numbers in stand-
ard form. Thus 34,900 = 3.49 xk 104 and 0.0012 = 1.2 x 10%.
*In using this rule, ‘‘first” is to be counted from the left; thus, in 3246, the digit 3 is “‘first.”’
[11]
Then
3.49 K 10! X 1.2 & 1073 = 3.49 & L.2 x 10!
= 4.19 < 10!
= 41.9
(ec) 6,230,000 x 0.00048. If Method ITI is used, think of this as about
50
6,000,000 x 100,000
which by “cancelling” zeros is easily found to be 3000. Use the C
and D scales to multiply 623 X 48, obtaining 299. Since the answer
is near 3000, it must be 2990.
(d) 6,230,000 « 0.00048. If Method III is used, write the numbers in
standard form.
6.23 X 10° x 4.8 X 1074 = 6.23 K 4.8 & 10?
= 29.9 X 10?
= 2.99 < 108
= 2990.
PROBLEM SET II
1. 24.9 XK 13.1 2. 37.2 X 16.6
3. 1728 XK 367 4. 1836 X 298
5. 0.0232 & 469 6. 0.0145 XK 543
7. 0.0715 X 3.48 8. 0.0627 K 4.34
9. 0.43 X 0.0027 10. 0.38 X 0.0042
11. 3,240 < 0.00039 12. 2.650 * 0.00047
13. 0.0061 X z 14, 0.0072 X «
15. 0.000,53 X 0.00623 16. 0.000,478 X 0.0069
DIVISION
In mathematics, division is the opposite or ¢nverse operation of multi-
plication. In using a slide rule this means that the process for multiplica-
tion is reversed. To help in understanding this statement, set the rule to
multiply 2 X 4 (see page 7). Notice the result 8 is found on the D scale
under 4 of the C scale. Now to divide 8 by 4 these steps are reversed.
First find 8 on the D scale, set 4 on the C scale over it, and read the result
2 on the D scale under the index of the C scale.
Think: From the length for 8 (on the D scale) subtract the length for 4
(on the C scale). The length for the difference, read on the D scale, is the
result, or quotient.
With this same setting you can read the quotient of 6 + 3, or 9 + 4.5,
and in fact all divisions of one number by another in which the result is 2.
Rule for Division: Set the divisor (on the C scale) opposite
the number to be divided (on the D scale). Read the result, or
quotient, on the D scale under the index of the C scale.
This may be shown in a diagram as follows. To find Q = x + y, set
Cilytd
Dji«!lQ
[ 12]
“Fen Seema ogg
©@ |OXAMPLFs:
(a) Find 63.4 + 3.29. The quotient must be near 20, since 60 + 3 = 20.
Set indicator on 63.4 of the D scale. Move the slide until 3.29 of the C seale is
under the hairline. Read the result 19.27 on the D seale at the C index.
(b) Find 26.4 + 47.7. Since 26.4 is near 25, and 47.7 is near 50, the quotient
must be roughly 25/50 = !4 = 0.5. Set 47.7 of C opposite 26.4 of D. using
the indicator to aid the eves. Read 0.553 on the PD seale at the C index,
(e) Find 371 + 0.00352. Write the problem in the form
3.71 X10 3.71
3.52% 10 = 3.52
Set the hairline over 37f on D. Move the slide so 352 on C is under the
hairline. At the index of C read 1054 on D. Start at the right of | and count
5 places, supplying zeros as needed, to obtain the result: 105,400.
xX 10°,
PROBLEM SET III
1. 47 + 29 2. 83+ 7
3. 75 + 92 4. 69 + 79
5. 137 + 513 6. 152 + 567
7. 0.049 + 0,023 8. 17.3 + 231
9. 924 + 26.3 10. 847 = 31.6
Ml. 0.0763 + 8.7 12. 0.0822 + 7.6
13. 428 + 0.34 14, 398 + 0.26
15. 564 + 0.047 16. 621 + 0.039
17, 8570 + 0.0219 18. 7630 + 0.0198,
19. 0.00146 + 32.8 20. 0.00231 + 41.3
CONTINUED PRODUCTS
Sometimes the product of three or more numbers must be found.
These “continued” products are easy to get on the slide rule. Below
is a general rule for continued products: aX 6 Xe Xd Ke...
Set hairline of indicator at a on D scale.
Move index of C scale under hairline.
Move hairline over 6 on the © scale.
Move index of C scale under hairline.
Move hairline over ¢ on the C scale.
Move index of C seale under hairline.
Yontinue moving hairline and index alternately until all
numbers have been set.
Read result under the hairline on the D scale.
¢ Examrpire: Multiply 38.2 x 1.65 * 8.9.
Estimate the result as follows: 40 X 1 x 10 = 400. The result should be,
very roughly, 400.
Setting the Scales: Set'left index of the C scale over 382 on the D scale. Find
165 on the C scale, and set the hairline on the indicator on it. The product of
382 x 165 could now be read under the hairline on the D scale, but this is
not necessary. Move the index on the slide under the hairline. In this example
[13]
if the left index is placed under the hairline, then 89 on the C scale falls
outside the D scale. Therefore move the right index under the hairline. Move
the hairline to 89 on the C scale and read the result (561) under it on the D
scale.
PROBLEM SET IV
1. 2.9 3.4 X 7.5
2. 17.38 * 43 X 9.2
3. 343 X 91.5 X 0.00532
4. 19 xX 407 X 0.0021
5. 13.5 X 709 x 0.567 x 0.97
COMBINED MULTIPLICATION AND DIVISION
Many problems call for both multiplication and division.
@ EXampLes:
(a) 42 X 37
65
First, set the division of 42 by 65; that is, set 65 on the ( scale opposite 42
on the D scale. The quotient, .646, need not be read. Move the hairline on
indicator to 37 on the C scale. Read the result 239 on the D scale under the
2
hairline. Since the fraction = is about equal to 5 the result is about two-
thirds of 37, or 23.9.
273 X 548
(b) 692 X 344°
Set 692 on the C scale opposite 273 on the 1) scale. Move the hairline to
548 on the C scale. Move the slide to set 344 on the C seale under the hairline:
Read the result .628 on the D scale under the C index.
In general, to do computations of the type bxdXfXho set the
rule to divide the first factor in the numerator @ by the first factor in the
denominator b, move the hairline to the next factor in the numerator e¢;
move the slide to set next factor in denominator, @d, under the hairline.
Continue moving hairline and slide alternately for other factors (e, f,
g, h, etc.). Read the result on the D scale. If there is one more factor in
the numerator than in the denominator, the result is under the hairline. If
the number of factors in numerator and denominator is the same, the
result is under the C index. Sometimes the slide must be moved so that
one index replaces the other. This statement assumes that up to this
point only the C and D scales are being used. Later sections will describe
how this operation may be avoided by the use of other scales.
® EXAMPLE:
2.2 X 2.4
@) BA
If the rule is set to divide 2.2 by 8.4, the hairline cannot be set: over 2.4 of
the C scale and at the same time remain on the rule. Therefore the hairline is
moved to the C index (opposite 262 on the D scale) and the slide is moved
end for end to the right (so that the left index falls under the hairline and
{14]
over 262 on the D scale). Then the hairline is moved to 2.4 on the C scale
and the result .63 is read on the D scale.
If the number of factors in the numerator exceeds the number in the
denominator by more than one, the numbers may be grouped, as shown
below. After the value of the group is worked out, it may be multiplied
by the other factors in the usual manner.
axbxe
( men ) xd Xe
PROBLEM SET V
1. 27 X 43 2. 5.17 X 1.25 & 9.33
19 4.3 X 6.77
3. 842 2.41 X 173 4. 1590 x 3.64 x 0.763
567 X 11.52 4.39 930
5. 0.0237 x 3970 X 32 X6.28 6. 231 X 58.6 X 4930
0.00029 x 186000 182.5 3770
7. 875 X 1414 X 2.01 8. 558 X 1145 x 633 x 809
661 X 35.9 417 X 757 X 354
9. 0.691 X 34.7 « 0.0561 10. 19.45 X 7.86 x 361 X 64.4
91,500 32.6 X 9.74
PROPORTION
Problems in proportion are very easy to solve. First notice that when
the index of the C scale is opposite 2 on the D scale, the ratio 1 :2 or 4
is at the same time set for all other opposite graduations” that is, 2 : 4,
or 3 :6, or 2.5 :5, or 3.2.:6.4, etc. It is true in general that for any
setting of the slide, many equal ratios are shown.
Fig. 10
Suppose one of the terms of a proportion is unknown. The proportion
can be writt f= wh i
en as; = — where a, b, and ¢, are known numbers and z is
to be found.
Rule: Set a on the C scale opposite b on the D scale. Under c
on the C scale read x on the D scale.
[15]
‘ :
This may be shown in a diagram as follows: C jase
D| ob] 2
© TXAMPLE:
Be 1 Ort
Find x if aa Set 3 on C opposite 4 on D. Under 5 on € read 6.67 on D.
: . bh os :
The proportion above could also be written aoe OF “inverted,” and
exactly the same rule could be used. Morever, if C and D are inter-
changed in the above rule, it will still hold if “under” is replaced by
“over.” It then reads as follows:
Set a on the 1 scale opposite 6 on the C scale. Over ¢ on the D scale
read x on the C scale.
Rule: In solving proportions, keep in mind that the position
of the numbers as set on the scales is the same as it is in the
A : . a €
proportion written in the form bod
Since one setting of the slide produces many equal ratios, it is pos-
sible to solve “continued” proportions by moving only the hairline.
® EXAMPLE:
Vind a, f, and g in the following.
GE eas OE eae a
12.5 30 f 75
Set 2.4 on C over 3.0 on D. Move the hairline:
over 12.5 on D, read a = 10 on C;
over 32 on C, read f = 40 on D;
over 7.50n D, read g 6 on C.
Sometimes the slide must be moved ‘‘end-for-end” because a reading
falls outside the body of the rule. Decimal points in continued propor-
tions are usually easy to locate by estimation (i. e., Method IT).
a
. . a ¢
Proportions can also be solved algebraically. Then a becomes
h 2
xz = —) and this may be computed as combined multiplication and
a
division.
In practical work it is often necessary to change a set of numbers
(like those in the table below) all to per cent.
® |exaMpvr: Find the missing per cents in the table.
Name Number Per cent |
Able | 76 | 124
Baker 21.4 15.1
Charlie 28.2 —_—
Dog 30.6 —
Fox _ 44.2 —
Total 142.0 100.0
[16]
To figure the per cents, each number (17.6, 21.4, ete.) must be divided by
the “base” or total, 142. Instead of re-setting the slide for each division, it is
easier to divide 1 by 142, then multiply the result by 17.6, by 21.4, etc. Note
that moving the hairline over 17.6 accomplishes the multiplication. This
method replaces five divisions by one division and five multiplications. Only
one setting of the slide is needed.
This problem can be viewed as a set of proportions, namely:
17.6 a 214 b 28.2 c
= -——; —~ = —~ ; ete.
142 100’ 142 100’ 142-100’
These can be written in the form
100 @ 68 ¢ @ _ ¢€
142° 17.6 214 282 306 44.2
Set 142 of Cover 1 of D. Move-hairline over 17.6 on C. Read 12.4% on D.
Then move hairline over 21.4 of C. Read 15.1% on D. Keep moving hairline
successively over other numbers in the table, reading per cents on D.
Finish the table, and check by adding the per cents. The sum should be
100%.
Hh other problems that require repeated divisions by the same number, a
similar procedure may be followed.
PROBLEM SET VI
8 2 14 35
12 23 18 13
3. = 4. ae
5, = 13.2 6 90.5 _ 3.42
* 425 ~~ 1.87 ‘““m 1.54
7 43.6 8 34.5 ¢
* 99.2 2550 * 73.8 2430
9 0.063 _ 34.1 10 0.0237
° "0.51 °° d * 0.0542 60.4
16 a 31
x 18.7 y
12. 338 ~ 192 ~ 345
14. or is
ail a ~ ro ~ F
[17]
MIXED PROBLEMS, SET VII
1. 143 K 0.387 2. 168 0.3824
3. 18.9 X 132 X 0.0481 4. 22.9 K 116 X 0.524
5. 832 + 6.41 6. 716 + 8.382
7 643 X 8.12 8 469 X 757
7 3.2 5 6.8
11. 1 + 3.48 12. 1 + 2.78
13. 29.8 X 4.87 14. 68.3 X 2.91
15. 79.1 XK 3.62 X 5.55 16. 93.2 X 22.1 X 0.625
17. 16.35 + 8.02 18. 14.62 + 7.03
11.95 X 9.12 16.28 X 5.37
ms 3.40 aM 4.60
1, 28h 2 22. 3.74
* 6.02 8.11 7.08 8.81
23. .0642 X 80.6 24. 0.0824 x 60.3
25. 0.0427 K 91.4 X 169 26. 0.0231 & 82.2 XK 182
27. 40.7 + 13.3 28. 61.3 + 15.1
7.75 X 0414 8.95 X 0.314
as 1.91 30. 2.09
PRACTICAL PROBLEMS, SET VIII
1. A stock that cost $27.60 a share paid a dividend of $2.25. What
rate of return (per cent) was this?
2. At 714¢ a square foot, how much will it cost to cover an area of
860 square feet?
3. Find the cost of carpet to cover a floor 13’ X 15’ at $9.85 per
square yard.
Bx ibn’ =? or 433 X 5 X 9.85 =?
4. A jet plane flew at the rate of 631 mi. per hr. Change this to feet
per second.
631 X 5280 _ 4
3600 _
5. It required 31% days to dig a ditch 520 feet long. At the same rate
how many days should be required to dig the next 370 feet?
6. In the fifth annual Pan American Road Race one car travelled the
1910 mile course in a little less than 17.7 hours. What was the
average speed?
7. If the voltage is 110 and the resistance is 73 ohms, what should the
current be?
E 110
I= Ro T= 73"
[18]
OV but
¢
PART 2. USE OF CERTAIN SPECIAL SCALES
THE CI AND DI SCALES
It should be understood that the use of the CI and DI scales does not
increase the power of the instrument to solve problems. In the hands
of an experienced computer, however, these scales are used to reduce
the number of settings or to avoid the awkwardness of certain settings.
In this way the speed can be increased and errors minimized.
The CI scale on the slide is just like the C scale except that it
increases from right to left. Starting at the right end, and reading toward
the left, one finds 2 of CI just above 5 of C, the 3 of CI is above 333 of
C, the 4 of CI is above 25 of C, ete. A small arrowhead (<) is placed at
the left of the numeral to show that the scale values increase toward
the left. The DI scale below the D scale is similar to the CI scale.
Fig. 11
bs en een Ee A EE RA AE cS PLA
25 333
Reciprocals. The reciprocal of a number a is 1 + @ or ~. The C
and CI scales (or the D and DI scales) may be used for finding
reciprocals.
Rule. When any number is set under the hairline on the C
scale its reciprocal is found under the hairline on the CI scale,
and conversely.
© EXAMPLES:
(a) Find 1/2.4. Set 2.4 on C. Read 417 directly above on CI.
(b) Find 1/60.5. Set 60.5 on C. Read .0165 directly above on CI. Or, set
60.5 on CI, read .0165 directly below on C.
(c) Find 1/. Place the hairline over + on D. Read 0.318 directly below
on DI,
Multiplication. When the C and D scales are used for multipli-
cation, it is sometimes necessary to change the slide ‘‘end-for-end.”’ This
happens if the factor on C is beyond the D scale. In division this never
happens and the answer can always be read. The CI scale can be used
to change a multiplication to an equivalent division. When multipli-
cation is done this way, the slide never has to be changed end-for-end,
Note that ; 1 1
P=axXb=a- (=> : ()).
Rule: The product of any two factors is equal to the one of
the factors divided by the reciprocal of the other factor.
{19 ]
< «
Tai aes Snnnoininienlncnmnenniaaal
This may be shown in diagrams as follows.
cij 6 | 1 cr| a | 1
DialP D/olP
or
© EXAMPLES:
(a) Find P = 2 X 8. Suppose | of C is placed over 2 on D. Then 8 on C
1
is beyond the D scale. Hence dimde 2 by 3 To do this set 8 on CI
over 2 on D. Under the index of CI read P = 16 on D.
Cr} 8 |
D|21P
\
Fig. 12
Cl ShinnlonRoinnbynBonwn hint hay nh Seba ying t Pi
ain im aur
1 : a 9 Mm
/ ™.
(b) Find 13.6 x 82. Set the hairline over 13.6 on D. Move the slide so
82 of CI is under the hairline. At the index of CI read 1115.
Division. The CI scale is useful in replacing a division by a multi-
plication. Since 5 = a X (1/b), any division may be done by multiplying
the numerator (or dividend) by the reciprocal of the denominator (or
divisor). This process may often be used to avoid settings in which the
slide projects far outside the rule. For example, consider Q = 16 + 8.
If 8 on C is set over 16 on D, the slide extends far out to the left. This
example may be done more easily by thinking of it as Q = 16 X (1/8).
Then the setting can be made as follows (see Fig. 12.):
D/| 16] 2
CI | 1 | 8
® EXAMPLES:
(a) Find 13.6 + 87.5. Consider this as 13.6 X 1/87.5. Set left index of the
C scale on 13.6 of the D seale. Move hairline to 87.5 on the CI scale.
Read the result, .155, on the D scale.
(b) Find 72.4 + 1.15. Consider this as 72.4 X 1/1.15. Set right index of
the C scale on 72.4 of the D scale. Move hairline to 1.15 on the CI
scale. Read 63.0 under the hairline on the D scale.
[ 20 ]
ecg «aR
Combined multiplication and division
By use of the CI scale, factors may be transferred from the numerator
to the denominator of a fraction (or vice-versa) in order to make the
rettings more readily.
® EXAMPLES:
(a) Find 13.