How to Use Dual Base Log Log Slide Rules (Pickett Form M-15)
FORM M -15
PRINTED IN U.S.A.
CHICAGO 5, ILLINOIS * SANTA BARBARA, CALIFORNIA
of Ws
Sree
bees:
by MAURICE L. HARTUNG
Professor of the
Teaching of Mathematics
THE UNIVERSITY OF CHICAGO
Price 50 Cents
PREFACE
Your DUAL-BASE LOG LOG Slide Rule is a powerful modern design that
has many features not found on traditional rules. These features make it easier
to learn how to use the rule. They make it easier to remember the methods of
solving problems as they arise in your work. They give you greater range on
some scales, and greater accuracy in the solution of many problems.
If you are familiar with traditional slide rules, don’t let the appearance of
the DUAL-BASE rule “scare” you. You will quickly recognize the basic features
of all slide rules as they have been improved in the design of this one. When
you have become familiar with your DUAL-BASE rule you will never again
be satisfied to use a traditional rule. The problems of today require modern
tools — better, faster, more accurate. Your DUAL-BASE rule is a modern tool.
A computer who must make many difficult calculations usually has a book
of tables of the elementary mathematical functions, or a slide rule, close at hand.
In many cases the slide rule is a very convenient substitute for a book of tables.
It is, however, much more than that, because by means of a few simple adjust-
ments the actual 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 other-
wise 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 underlying the operations. Some explain
in detail how to solve many different types of problems — for example, various
cases which arise in solving triangles by trigonometry. 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 in-
tended to be a textbook or workbook. Relatively few exercises are included,
and the answers of these (to slide rule accuracy) are given immediately so that
learning may proceed rapidly, by self-correction if necessary. Any textbook on
mathematics which contains problems suitable for slide rule calculation, and
their answers, will provide additional practice.
Some of the special scales described in this manual may not be available
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 calcu-
lations. Pickett Slide Rules are available with all of the special scales shown
in this manual.
TABLE OF CONTENTS
Part 1—SLIDE RULE OPERATION PAGE
INtrOGURMEREEET Os cout ce Sere ee a cag ouster aitke ee ales 8% 5
How to Read the Scales............. 0.000 e eee eeeee 6
‘Multiplicotiemitan.s. siiachacal privicval eweiesuiariest. 9
Division aie i tren sie ecid Ao. ge slow tania sbeveis carrer ¢ « 10
Decimal Point Location ..............00 00 ee eee eens 11
Continued Products ............. 020 ce eee eee eens 13
Combined Multiplication and Division................ 13
Proportngimiene¢ <8 of Josaosawd. want ania aadpawds « « 15
PART 2—USE OF CERTAIN SPECIAL SCALES
The Cisdnd Dt, Scalesiiesecjciad sad 2 dadi enisawis.. 17
The CFz and DFz Scales... 2... 0565 ee ce eee ee eee 19
TG Cian so kc 8 cs oe bw 8 wwe iw wea Caw sev ww asd 20
The GF/M and DF/M Scales... . ood cece cece ees ese 21
The \/ Scales: Square Roots and Squares............. 21
The \/ Scales: Cube Roots and Cubes................ 23
The L Scale: Logarithms. ... 2.0... se. ee ee eee ee 26
The S, T, and ST Scales: Trigonometry..........i..... 27
PART 3—ELEMENTARY VECTOR METHODS
Complex Numbers and Vectors. ............--+-+005- 36
Illustrative Applied Problems ...................... 43
PART 4— USE OF LOG LOG SCALES
Reading the-Scales..........ebaxi. T.alaaomi. ie.e2ms... 45
Fundamental Relationships ................2+e+e005 49
Finding Logarithms, Base E................-0eeee ees 50
Finding Logarithms, any Base................+++055 52
Finding Powers of Biss sik: sess cn e's olga se 0 oe 53
Finding Powers of any Base...............000eeees 54
Roots and Common Fractional Exponents............. 58
Solving Exponential Equations. ..............000005: 60
Logarithms of Complex Numbers................+-> 62
Readings Beyond the Scales..............+--eeeeees 63
Illustrative Applied Problems. .............----+0055 64
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PART 5— PROBLEMS AND ANSWERS
Exercises for Practice... 2.0... ce ee ee ee ees 65
DHSWERE cn ncacnsnc cr TRO HH MEROOMEE OO Ne eo 66
PART 6—HYPERBOLIC FUNCTIONS OF REAL VARIABLES
Evaluating the Functions. ..... 0.0... 00 cee eee eens 67
Computations involving Hyperbolic Functions.......... 68
Large and Small values of the Argument............. 69
PART 7—HYPERBOLIC FUNCTIONS OF COMPLEX ARGUMENTS
Changing sinh z from Component to Exponential Form. . .70
Alternative Methods for sinh z............-555-000> 75
Changing cosh z from Component to Exponential Form. .76
Alternative Methods for cosh z.......... 00 ee eeeee 78
Changing tanh z from Component to Exponential Form. .79
Wustrative Applied Problems. .............002e sees 80
PART 8—INVERSE HYPERBOLIC FUNCTIONS OF
COMPLEX ARGUMENTS
Finding z for sinh z = P/? 2.2... eee 83
Finding z for cosh z = p/? ...........-- SIRENS ¢ 84
Finding z for tanh z = p/0 2... eee ee eee 85
Hlustrative Applied Problem...............-5+-5005 86
PART 9—CIRCULAR FUNCTIONS OF COMPLEX ARGUMENTS. . .87
PART 10—One-Step solution of “Two sides and
included angle triangle” problem through
use of Double T Scale... 1.0... ee ees 90
{2}
ante
va
VECTOR TYPE LOG LOG DUAL BASE
EXPONENTIAL SPEED RULE
Recommended
for Upper Lavel
CORLEGE and Supenor
HIGH SCHOOL
Students
For advanced Mathematics, Science, Electronics and Engi-
neering.
{3 ]
PART 1. SLIDE RULE OPERATION
INTRODUCTION
The slide rule is a fairly simple tool by means of which answers to involved
mathematical problems can be easily obtained. 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.
ae The slide rule consists of three parts: (1) the stator (upper and lower
bars); (2) the slide; (3) the cursor or indicator. The scales on the bars 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 in adjusting the slide.
y STATOR
Fig. 1 SUDE HAIRLINE ~~
STATOR CURSOR OR | INDICATOR
\ —
Each scale is named by a letrer (A, B, C, D, L, S, T) of other symbol at
the left end.
The table below shows some of the mathematical operations which can be
done easily and quickly with an ordinary slide rule.
OPERATIONS INVERSE OPERATIONS
Multiplying two or more numbers| Dividing one number by another
Squaring a number Finding the square root of a number
Cubing a number Finding the cube root of a number
‘ Finding the logarithm of a number! Finding a number whose logarithm is
known
v Finding the sine, cosine, or tangent Finding an angle whose sine, cosine, or :
of an angle tangent is known |
Various combinations of these operations (such as multiplying two numbers
and then finding the square root of the result) are also easily done. Numbers
i can be added or subtracted with an ordinary slide cule, but it is usually easier
i to do these operations by arithmetic. *
| In order to use a slide rule, a computer must know: (1) how to read the
j scales; (2) how to “‘set” the slide and indicator for each operation to be
done; and (3) how to determine the decimal point in the result. |
“By putting special scales on a slide tule, these and certain other operations much more difficult
than those shown in the table above can be done easily.
[5]
LEFT INDEX
LEFT INDEX 7 54
HOW TO READ THE SCALES
The scale labeled C (on the slide) and the scale D (on the stator bar) 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
scale the numerals are printed above the corresponding preduations. A line
labeled 1 at the left end is called the /eff index. A line labeled 1 at the right
end is called the right index.
s ECU ERI DUTYEAQUT TOE SETI LUE! SUOUNG DOWEL UUTT OULU QCRULRD UGA AOA OLY LL
| | | |
PRIMARY GRADUATIONS
Fig. 2
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 secondary
graduations. (On 10 inch slide rules these lines are labeled with smaller
numerals 1, 2, 3,-etc. 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 graduations because it would
crowd the numerals too much.
; MOLY ULL LL MM AA ERE UO UO AC
SECONDARY GRADUATIONS
Fig. 3.
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 17 is located, the 1 at the left
index represents the 1 in 17, The 7th secondary graduation represents the 7.
When 34 is to be located, look first for primary graduation 5, 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 G inch slide rule chan on a 10 inch rule. For this
reason a separate explanation must be given for each.
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RIGHT INDEX
RIGHT INDEX
fcc COTTE
D 1 2 3 4
Tertiary graduations on 10 inch rules.
The space between each secondary graduation at the left end of the rule
(over to primary graduation 2) ts 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 between the secondary
graduations are separated into five parts. Finally, the still smaller spaces be-
tween the secondary graduations at the right of 4 are separated into only
two parts,
To find 173 on the D scale, look for primary division 1 (the left index),
then for secondary division 7 (numbered) then for smaller subdivision 3 (not
numbered, but found as the 3rd very short graduation to the right of the
longer graduation for 7).
149 73 246 247
Fig. 4 :
/
UE Le LUA
LEFT INDEX
c
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 smal! graduation after
it. The smallest spaces in this part of the scale are fifths. Since 3 = +5, then
the third graduation, marking three fifths, is at the same point as six tenths
would be.
Tertiary graduations on 6 inch rules.
The space between each secondary graduation at the left end of the rule
(over to primary graduation 2) is separated into five parts. In the middle of
the rule, Feeween the primary graduations 2 and 5, the smaller spaces between
the secondary graduations are separated into two parts. Finally, the still
smaller spaces between the secondary graduations at the right of 5 are not
subdivided.
To find 170 on the D scale, look for the primary division 1 (the left index),
then for the secondary division 7 which is the 7th secondary graduation.
Fevvadaecaluveiduvatlieylissstuoseuadeeiat |b eee Uy Ee Leeder bebe data Latadetalefittail dR vveda dF veeteseafineatie acai
fe RIGHT INDEX
146 147 170 383 385
Fig. 5
{7]
ayy AC
D
To find 146, look for the primary graduation 1, then for the 4th secondary
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 fifths. Since 3/5 = 6/10,
then the third graduation, marking three fifths, is at the same point as six
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 small space is mentally “‘split’’ in half.
The number 385 is found by locating primary graduation 3 and then
secondary graduation 8 (the 8th long graduation after 3). Following this, one
observes that between secondary graduations 8 and 9 there ts one short mark.
Think of chis as the ‘5 cenths’’ mark, which represents 385. The location of
383 can be found approximately by mentally ‘splitting’ the space becween
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.
D Cc B A OF E
Jog sl gj ng ec TERT
H G !
TULL UM TP UD LUM Uo OU UO GMO COuOU UL
Fig. 6
A: 193 F: 206
B: 119 G: 465
Cc: 110 H: 402
D: 101 I: 694
E: 223 J: 987
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 54 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 de-
gtee of accuracy is sufficient. When greater accuracy 1s desired, a ten inch
slide rule is generally used.
{8}
“alent:
<
MULTIPLICATION
Numbers that are to be multiplied are called factors. The result is called
the product. Thus, in the statement 6 X 7 = 42, the numbers 6 and 7 are
factors, and 42 is the product.
EXAMPLE: Muluply 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 len rth, measured by the D scale, is 6.
LEFT iNDEX
“e!" SCALE
on sgaent LENGTH OF FACTOR “3 ON C1" SCALE
—-
|
1 2 3 4 5 ie t ad { 7m A
ug nga og, png LI a agai eh uk
|
I
LENGTH OF “2” PLUS LENGTH OF "3 1S LENGTH OF PRODUCT ‘6 ON “’0’ SCALE
LEFT INDEX
ON 0" SCALE
pps
LENGTH OF FACTOR ‘2 ON “D’’ SCALE
EXAMPLE: Multiply 4 X 2 Fig. 7
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, ts 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.
EXAMPLE. Multiply 2.34 K 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 XK 368.
Set the Scales: Set the left index of the C scale on 234 of che D scale. Find
368 on the C scale and read product 861 on the D scale.
Think: The length for 234 plus the length for 368 will be the length for the
product. This length 1s measured on the D scale. Since we already knew the
result was somewhere near 80, the product must be 86.1, approximately.
EXAMPLE: Multiply 28.3 X 5.46
Noce first chat rhe result will be abouc che same as 40 X 5, or 150. Nore
also chat if the left index of the C seale is set over 283 on the D scale, and
546 is then found on the C scale, the slide projects so far co the right of the
rule that the D scale is no longer below the 546. When this happens, the
other index of the C scale must be used. That ts, set the right index on the C
scale over 283 on the D scale. Find 546 on the C scale and below ic read
the product on the D scale. The product ts 154.5.
These examples illustrate how in srmple problems the decimal point can be
placed by use of an estimate.
[9]
PROBLEMS ANSWERS
1. 15 X 3.7 53.5
2. 280 X 0.34 95.2
3. 753 X 89.1 67,100
4. 9.54 X 16.7 159.3
5. 0.0215 X 3.79 0.0815
DIVISION
In mathematics, division is the opposite or /nverse operation of multiplica-
tion. In using a slide rule this means that the process for multiplication is
reversed. To help in understanding this statement, set the rule to multiply
2X 4 (see page 9). 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 + 4.8,
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.
EXAMPLES:
(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 scale is
under the hairline. Read the result 19.27 on the D scale 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 eyes. Read 0.553 on the D scale at the C index.
PROBLEMS ANSWERS
1. 838+ 7 11.86
2. 75 + 92 0.815
3. 137 + 513 0.267
4. 17.3 + 231 0.0749
5. 8570 + .0219 391,000
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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 digits 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
.678, or six-hundred-seventy-three thousandths, is a decimal fraction. An-
other example is .000465, In this number three zeros are written to show
where the decimal point is located. One way to describe such a number is
to tell how many zeros are written to the right of the decimal point before
the first non-zero digit occurs.
In scientific 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 not be counted unless they are (a) at the right of the decimal point,
(b) before or at the /eft of the first non-zero digit, and (c) are not between
other digits. The number 0.000408 will be said co have 3 zeros (that is, the
number of zeros between the decimal point and the 4).
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 ‘‘reason-
able’ 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 be-
longs between the 3’s, since 48 m.p.h. is too smal!, and 4833 m.p.h. 1s 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.
Another commonly used method of locating the decimal point is by estima-
tion 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 the decimal point
is to be determined. By rounding off the factors to 133.0 X 10.0, one ob-
tains 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 1655.
In scientific work numbers are often expressed in standard form. For exam-
ple, 428 can be written 4.28 X 10?, and 0.00395 can be written as 3.95 X 107%.
When a number is witten 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 tight 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.
*Only positive real numbers are being considered in this discussion.
[11]
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 negative
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 decimal 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 ex-
ponent of 10. If the original decimal point is toward the right, the char-
acteristic is positive (+). If the original decimal point is toward the left,
the characteristic is negative (—). Indicate that the result of (a) is to be
multiplied by 10 with the exponent thus determined in (b).
Rule. To determine the decimal point, first express the numbers in
standard form. Carry out the indicated operations of multiplication or
division, using the laws of exponen to combine the exponents until a
single power of 10 is indicated. If desired, write out the resultin number,
using the final exponent of 10 to determine how far, and in what direc-
tion, the decimal point in the coefficient should be moved.
CONTINUED PRODUCTS
Sometimes the product of three or more numbers must be found. These
“continued” products are easy to get on the slide rule.
ExAMPLe: Multiply 38.2 x 1.63 X 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 Aairline on the indicator on it.** Move the index
on the slide under the hairline. In this example if the /eft 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.
Below is a general rule for continued products: 4X 6X¢XdXer--
Set hairline of indicator at 24 on D scale.
Move index of C scale under hairline.
Move hairline over 6 on the C scale.
EXAMPLES:
Number in
Number standard form. Characteristic
(a) 5,790,000 5.79 X 108 6
(b) 0.000283 2.83 X 107* —4
(c) 44 4.4 X 10! 1
(d) 0.623 6.23 X 107! —1
(e) 8.15 8.15 X 10° 0
(fF) 461,328 4.61328 X 10° 5
(g) 0.0000005371 5.371 X 1077 —7
(h) 0.0306 3.06 X 107 —2
(i} 80.07 8.007 X 10! 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 is toward the left. This con-
i ti application of the rule may be verified by studying the examples given
above.
Consider now the calculation of 5,790,000 X 0.000283. From examples
(a) and (b) above, this can be written 5.79 X 10° X 2.83 x 10-4, or’ by
changing order and combining the exponents of 10, as 5.79 X 2.83 X 102.
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 X 10. Hence, 5.79 X 2.83 X 10?
= 1.6389 x 10' K 10? = 1.639 X 10° = 1639. If, however, one has
5,790,000 + 0.000283, the use of standard form yields
3.79 X 108
ee 6—(-4) —
283 X 10 2.04 10 2.04 X 10.19
In scientific work the result would be left in this form, but for popular con-
sumption it would be written as 20,400,000,000. The general rule is as follows.
*In using this rule, “‘firse’’ is to be couated from the left; chus, in 3246, che digic 3 is “first.”
[12]
Move index of C scale under hairline.
Move hairline over ¢ on the C scale.
Move index of C scale under hairline.
Continue moving hairline and index alternately until all mumbers have
been set.
Read result under the hairline on the D scale.
PROBLEMS ANSWERS
1. 2.9 X 34 X 7.5 73.9
2. 17.38 X 43 X 9.2 6,840
3. 343 X 91.5 X 0.00532 167
+. 19 K 407 0.0021 16.24
5. 13.5 & 709 X 0.567 X 0.97 5260
COMBINED MULTIPLICATION AND DIVISION
Many problems call for both multiplication and division.
E 42 X 37
XAMPLE: 65
*See any textbook on elementary algebra. The theory of exponents and the rules of operation
with signed numbers are both involved in a complete treatment of this topic. In chis manual it
is assumed chac the reader is familiar with this theory.
**The product of 382 X 165 could now be read undet the hairline on the D scale, but this is
not necessary. :
{13}
First, set the division of 42 by 65; that is, set 65 on the C scale opposite 42
on the D scale. * Move the hairline on indicator to 37 on the C scale. Read
42
the result 239 on the D scale under the hairline. Since the fraction ra is about
equal to A the resule is about two-thirds of 37, or 23.9.
273 XK 548
EXAMPLE: 692 x 344
Set 692 on the C scale opposite 273 on the D scale. Move the hairline to
548 on the C scale. Move the slide to set 344 on the C scale under the hairline.
Read the result .628 on the D scale under the C index.
In general, to do computations of the type ; = 5 : rx ; : set the rule
to divide the first factor in the numerator a by the first factor in the de-
nominator 6, move the hairline to the next factor in the numerator c; move
the slide to set next factor in denominator, d, under the hairline. Continue
moving hairline and slide alternately for other factors (¢, f, g, 4, 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.**
2.2 X 2.4
EXAMPLE: 34
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 /eft index falls under che hairline and
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 de-
nominaror 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.
PROBLEMS BREWERS
1. ee 61.1
ee 5 ed a
3. 842 x a EI 53.7
4. 1590 X 364 X 0.768 1.081
*The quotient, .646, need not be read,
**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,
{14}
5, 0.0237 K 3970 X 32 X 6.28 351
0.00029 x 186000
6, 231 X 58.6 X 4930 97.0
182.5 X 3770
7, 875 X 1414 X 2.01 104.8
661 & 35.9
g, 558 X 1145 X 633 x 809 2930
417 X 757 X 354
g. 0.691 X 34.7 X 0.0561 0.0000147
91,500 or 1.47 X 1075
10. 19.45 x 7.86 X 361 x 64.4 11,190
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 3 is at the
same time set for all other opposite graduations, that is, 2:4, of 3:6, or
25-5, 0r3.2:64, etc. It is true in general that for any setting the numbers
for all pairs of opposite gradustions have the same ratio. Suppose one of the
; aie
terms of a proportion is unknown. The proportion can be written as 7 = 3,
where a, 6, and c, are known numbers and x is to be found.
Rule: Set a on the C scale opposite 6 on the D scale. Under c on the C
scale read x on the D scale.
. 3 _5
EXAMPLE: Find x if gy
Set 3 on C opposite 4 on D. Under 5 on C read 6.67 on D.
a x aac ”
The proportion above could also be written zoe ot inverted,” and
exactly the same rule could be used. Moreover, if C and D are interchanged
in the above rule, it will still hold if “under” is replaced by “over.” It then
reads as follows:
Set a on the D scale opposite b 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 itis in the proportion written
: 4
in the form aa
{15]
be PART 2. USE OF CERTAIN SPECIAL SCALES
a
Proportions can also be solved algebraically. Then ; = £ becomes x=
. . bene 3 3 THE CI AND DI SCALES
and this may be computed as combined multiplication and division.
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
PROBLEMS ANSWERS 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
13.2 and errors minimized.
Ls 2 ged Pe 300. i The CI scale on the slide 1s a C scale which increases from right to left. It
42.5 187 : may be used for finding reciprocals. When any number is set under the hair-
2 90.5 3.42 40.7 j line on the C scale its reciprocal is found under the hairline on the CI scale,
x 154 and conversely.
EXAMPLES:
3 ee _ aa 1247. (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
4, 0.063 _ 34.1 276. 60.5 on CI, read .0165 directly below on C.
wo = ® The CI scale is useful in replacing a division by a multiplication. Since
5. = = = 65.7 rc a X 1/4, 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.
EXAMPLES:
(a) Find 13.6 + 87.5. Consider this as 13.6 X 1/87.5. Set left index of the
CG scale on 13.6 of the D scale. 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.
An important use of the CI scale occurs in problems of the following type.
EXAMPLE: Find 13.6 This is the same as 13.6 X (1/2.79).
4.13 X 2.79 4.13
Set 4.13 on the C scale opposite 13.6 on the D scale. Move hairline to 2.79
on the CI scale, and read the result, 1.180, on the D scale.
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 settings more
teadily. Also, it is sometimes easier to get a X b by setting the hairline on a,
pulling 6 on the CI scale under che hairline, and reading the result on the D
scale under the index.
The DI scale (inverted D scale) below the D scale corresponds to the CI
scale on the slide. Thus the D and DI scales together represent rectprocals.
The DI scale has several important uses, of which the following ts representa-
tive.
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[16]
Finally, these scales are useful in changing radians to degrees and con-
versely. Since w radians = 180 degrees, the relationship may be written as a
7 ofa #
180° x)= «180
Rule: Set 180 of C opposite on D. To convert radians to degrees, move
indicator to r (the number of radians) on DF, read d (the number of de-
grees) on CF; to convert degrees to radians, move indicator to ¢ on CF,
read r on DF.
There are also other convenient settings as suggested by the proportion.
Thus one can set the ratio 7/180 on the CF and DF scales and find che result
from the C and D scales.
EXAMPLES:
(a) The numbers 1, 2, and 7.64 are the measures of three angles in radians.
Convert to degrees. Set 180 of Con 7 of D. Move indicator to 1 on DF, read
57.3° on CF, Move indicator over 2 of DF, read 114.6°. Move indicator to
7.64 of DF. Read 437° on CF.
(b) Convert 36° and 83.2° to radians. Use the same setting as in (a) above.
Locate 36 on CF. Read 0.628 radians on DF. Locate 83.2 on CF. Read 1.45
radians on DF.
: r
tion — =
proportion ~
PROBLEMS ANSWERS
1. 1414 x 7.79 11.02
2. 2.14 x 576 123.3
3 84.5 X 7.59
eRe 17.43
4, 265X « 8.33
5, .1955 X 23.7
50.7 Mr 0291
THE CIF SCALE
Like the other special scales the CIF scale does not increase the power of
the instrument to solve problems. It is used to reduce the number of settings
or to avord the awkwardness of certain settings. In this way the speed can be
increased and errors minimized,
The CIF scale is a folded CI scale. Its relationship to the CF and DF scales
is the same as the relation of the CI scale to the C and D scales.
EXAMPLES:
(a) Find 68.2 X 1.43. Set the indicator on 68.2 of the D scale. Observe
that if the left index is moved to the hairline the slide will project far to the
right. Hence merely move 14.3 on CI under the hairline and read the result
97.5 on D at the C index.
(b) Find 2.07 * 8.4 X 16.1. Set indicator on 2.07 on C. Move slide until
8.4 on CI is under hairline. Move hairline to 16.1 on C. Read 280 on D under
hairline. Or, set the index of CF on 8.4 of DF. Move indicator to 16.1 on CF,
chen move slide until 2.07 on CIF is under hairline, Read 280 on DF above the
index of CF. Or set 16.1 on CI opposite 8.40n D. Move indicator to 2.07 on C,
and read 280 on D. Although several other methods are possible, the first
method given is preferable.
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9 ime pemetaie gods:
THE CF/M AND DE/M SCALES
The scales CF/M and DF/M are a special feature of PICKETT DUAL BASE
LOG LOG SLIDE RULES. Later sections (See Part 4) will explain how the
Log Log scales are used. For the present, we only need to say that on DUAL
BASE rules, logarithms to base 10 may be read from the ordinary C or D
scales, and “Natural” logarithms (to base ¢) may be read directly (with the
same setting of the indicator) from these special CF/M and DF/M scales.
The CF/M and DF/M scales are folded at 1/M=2.30, where the modulus
M=log,,¢. The symbol /M is used to distinguish these scales from the ordinary
CF and DF scales which are folded x. The index | of CF/M and DF/M is about
two-thirds of the length of the rule from the left end.
Although these scales have a special purpose, they may be used in ordinary
multiplication, division, and other calculations in exactly the same way that
the CF/r and DF/z scales are used. This means, in effect, that PICKETT
DUAL BASE LOG LOG SLIDE RULES not only have ordinary C and D scales
on both sides of the rule, but also have a set of folded scales on each side.
EXAMPLES: (a) 19.26.38. Set left index of C on 19.2 of D. Note that
6.38 on C falls outside the D scale. Hence, move the indicator to 6.38 on the
CF/M scale, and read the result 122.5 on the DF/M scale. Observe that the
index of the CF/M scale is automatically set at 19.2 of the DF/M scale.
8.39 X9.6
(b) Find soe
not be moved to 9.65 of C, but it can be moved to this setting on CF/M and
the result, 14.15, read on the DF/M scale. Or the entire calculation may be
done on the CF/M and DF/M scales.
Set 5.72 on C opposite 8.39 on D. The indicator can-
THE V SCALES: Square Roots and Squares
When a number is multiplied by itself the result is called the sguvare of the
number. Thus 25 or 5X5 is the square of 5. The factor 5 is called the square
root of 25. Similarly, since 12.25 =3.5 X3.5, the number 12.25 is called the
square of 3.5; also 3.5 is called the square root of 12.25. Squares and square
roots are easily found on a slide rule.
Square Root. Just below the D scale is another scale marked with the square
root symbol, V.
Rule, The square root of any number located on the D scale is found
directly below it on the V scale.
EXAMPLES: Find V4. Place the hairline of the indicator over 4 on the
D scale. The square xoot, 2, is read directly below. Similarly, the square root
of 9 (or V9) is 3, found on the V scale directly below the 9 on the D scale.
{21]
Reading the Scales. The square root scale directly below the D scale is an
enlargement of the D scale itself. The D scale has been “stretched” to double
its former length. Because of this the square root scale seems to be cut off
or to end with the square root of 10, which is about 3.16. To find the square
root of numbers greater than 10 the bottom V scale is used. This is really
the rest of the stretched D scale. The small figure 2 near the left end is placed
beside the mark for 3.2, and the number 4 is found nearly two inches farther
to the right. In fact, if 16 is located on the D scale, the square root of 16, or
4, is directly below it on the bottom scale of the rule.
In general, the square root of a number between 1 and 10 is found on the
upper square root scale, The square root of a number between 10 and 100
is found on the lower square root scale. If the number has an odd number
of digits or zeros (1, 3, 5, 7,...), the upper V scale is used. If the number
has an even number of digits or zeros (2, 4, 6, 8,...), the lower V scale is
used. The first three (or in some cases even four) figures of a number may be
set on the D scale, and the first three (or four) figures ‘of the square root are
read directly from the proper square root scale.
The table beiow shows the number of digits or zeros in the number N and
its square root.
ZEROS of DIGITS
U LIU LIU L;U; LIU LIV LIU LIU L
N | 7or6/[Sor4}3o0r2/ 1 lor 2]3 or 4j5 or 6G] 7 or 8 | etc.
V/N| 3 2 1 |O} 0] 1 2 3 4 | ete,
oQ
The above table is reproduced on some models of Pickett Slide Rules.
This shows that numbers from 1 up to 100 have one digit in the square
root; numbers from 100 up to 10,000 have two digits in the square root, etc.
Numbers which