How to Use Log Log Slide Rules (Pickett Form M-14; reference for Models 800, 803, 300, 600 and N3, with the 803 supplement)
FORM M-14
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CHICAGO 5, ILLINOIS « ALHAMBRA, CALIFORNIA
Sees RINE NN ORNS PEE SCOR TEAL
by MAURICE L. HARTUNG
Associate Professor of the
Teaching of Mathematics
THE UNIVERSITY OF CHICAGO
Price 50 Cents
PREFACE
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 resule 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 & Eckel Slide Rules are available with all of the special scales
shown in this manual.
COPYRIGHT 1953 * PICKETT & ECKEL INC * CHICAGO § ILL
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TABLE OF CONTENTS
PART 1—SLIDE RULE OPERATION PAGE
Introduction... 00.0... . 2. et tenes 3
How to Read the Scales... ......---- 0: eer ee cts 4
Multiplication. ........ 0.0.0.0. eee et ree eee 7
Division... 2... eee eee eens 8
Decimal Point Location. ..........-0 00: eee eee 9
Continued Products. .........0- 002s ee eer teense 11
Combined Multiplication and Division. .......-.....-.-11
Proportion... 0.0.0.6... cece eee eee erent eens 13
Special Graduations —Circular Measure.....-.-------- 14
PART 2—USE OF CERTAIN SPECIAL SCALES
The Cl and D1 Scales... 0.0... 00... ee 15
The CFz and DFr Scales... 0.0... ce ees 17
The CIF Scale... 1... ee tenes 18
The A and B Scales: Square Roots and Squares... ........- 19
The K Scale: Cube Roots & Cubes............ 02-0055 21
Combined Operations. .......... 060 e eee e reer e eee 23
The L Scale: Logarithms. ...........- 2002 ee eens 24
The S, T, and ST Scales: Trigonometry..........---.+-- 25
PART 3—ELEMENTARY VECTOR METHODS
Complex Numbers and Vectors..........--+++-+++-- 32
Illustrative Applied Problems. ... . Bg 6 5a oe w rosaries 37
PART 4—USE OF LOG LOG SCALES
Reading the Scales.......--..-- 202 eee cece eee 39
Fundamental Relationships ..........-. 6-000 er sees 45
Finding Logarithms, Base E.......----- +2 e eee eee 46
Finding Logarithms, any Base. ..... 6... 260. eee eee 48
Finding Powers of E.... 1... 2-25 - ee eee ee eee reer 49
Finding Powers of any Base.......-..----+ sere erees 49
Roots and Common Fractional Expontnes...........--: 55
Solving Exponential Equations.............--++-++:- 56
Logarithms of Complex Numbers... .......--- 0050055 58
Readings Beyond the Scales. ..........-.--0 eee eeeee 59
Illustrative Applied Problems...........-.---++-++5- 60
PART 5—PROBLEMS AND ANSWERS
Exercises for Practice. 2.0.00 ec eee eee 61
AMSWErs. 0 ee eee eee eee ene ene 62
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PART I. 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.
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 re
ading the scales and in adjusting the slide.
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Each scale is named by a letter
the left end.
(A, B, C, D, L, S, T) or other symbol at
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
Squaring a number
Cubing a number
Finding the logarithm of a number
Finding the sine, cosine, or tangent
Dividing one number by another
Finding the square root of a number
Finding the cube root of a number
Finding a number whose logarithm is
known
Finding an angle whose sine, cosine, or
tangent is known
of an angle
Various combinations of these operations (such as multiplying two numbers
and then finding the square root of the result) are also easily done. Numbers
can be added or subtracted with an
ordinary slide rule, but it is usually easier
to do these operations by arithmetic.*
In order to use a slide rule, a computer must know: (1) how co read the
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 rule, these and certain other operations much more difficult
~ than those shown in the table above can be done easily.
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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 (i, 2, 3, etc.)
printed just below them. These lines are called primary graduations. On the C
scale the numerals are printed above the corresponding ae ee: A line
labeled 1 at the left end is called the /eft index. A line labeled 1 at the right
end is called the right index.
Tertiary graduations on 10 inch rules.
The space between each secondary graduation at the left end of the rule
(over to primary graduation 2) is separated into ten parts, but these shortese
graduation marks are nor numbered. In the middle part of the rule. between
the primary graduations 2 and 4, the smaller spaces between che 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 patts.
To find 173 on the D scale, look for primary division 1 (che 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).
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LEFT INDEX | | | | RIGHT INDEX
PRIMARY GRADUATIONS
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LEFT INDEX
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.
Fig. 4
Similarly, 149 is found as the 9th small graduation mark to the right of
the tch 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 $ = 75. then
the thitd graduation, marking three fifths, is at the same point as ix tenths
would be.
p [HHL init a LL RUUD CHUL PO IL LLY CLO L COU CS Tertiary graduations on 6 inch rules.
LEFT INDEX 7 54 RIGHT INDEX The space between each secondary graduation at the left end of the rule
SECONDARY GRADUATIONS (over to primary graduation 2) is separated into five parts. In the middle of
the rule, between the primary graduations 2 and 5, the smaller spaces between
the secondary graduations are separated into two parts. Finally, che still
Fig. 3. smaller spaces between the secondary graduations at the right of 5 are not
subdivided.
; ; oo , To find 170 on the D scale, look for the primary division 1 (the left index),
When a number is to be located on the D scale, the first digit is located by then for the secondary division 7 which is the 7th secondary graduation.
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 54 is to be located, look first for primary graduation 5, and chen for © dritritinin iqliitenerude itil Mustat Lt vvtatifiet dean rete ta® catia Ter atl featuPantued
secondary graduation 4 in the space immediately to the right. —_ wm | RIGHT INDEX
There are further sub-divisions, or sertiary graduations, on all slide rules. ~ Wey TERT’ (DEX bod
The meaning of these graduations is slightly different ac different parts of the )“#/» ; 146 147 170 a03 ges
scale. Ii is also different on a G inch slide rule than on a 10 inch rule, For this //s.ue RULE a
reason a separate éxplanation must be given for gach. wd Fig.
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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 chis part of the scale are fifths. Since 3/5 = 6/10
then the third graduation, marking three fifths, is at the same poine as tise
tenths would be.
The number 147 would be half of a smail 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 beeween secondary graduations 8 and 9 there is one short mark.
Think of this as the ‘5 tenths’? mark, which represents 385. The location of
383 can be found approximately by mentally “‘splicting’’ the space between
380 and 385 into fifths, and estimating where the 3rd ‘fifths’ mark would be
placed. It would be just a lictle to the right of halfway berween 380 and 385.
On the scale below are some sample readings.
D Cc B
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A: 195 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 writtng 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 firsc two or three digits of a number
need to be used to get an approximate result which ts accurate enough for
pracuical 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-
gree of accuracy is sufficient. When greater accuracy 1s desired, a ten inch
slide rule ts generally used.
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LEFT INDEX
ON “D" SCALE
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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 ts 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 length, measured by the D scale, is 6.
LEFT INDEX
ON "'C’" SCALE
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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, 1s 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 X 36.8
Estimate the result; First note that the resulc 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 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 is 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
Note first that the resule will be about 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
tule that the D scale is no longer below the 546. When this happens, the
other index of the C scale must 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 scale. The product is 154.5.
These examples illustrate how in simple problems the decimal point can be
placed by use of an estimate.
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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 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 resulc 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.
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 che 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 quotienc
must be roughly 25/50 = % = 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. 88+ 7 11.86
2. 75 + 92 0.815
3. 1387 + 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
.673, or six-hundred-seventy-three thousandths, is a decimal fraction. An-
other example is .000465. In this number three zeros ate 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 berween
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 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.
Another commonly used method of locating the decimal point is by estima-
tion or approximation. For example, when the slide rule ts 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 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.
*Only positive real numbers are being considered in this discussion.
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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 Aecimal 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
ind 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).
EXAMPLES:
Number in
Number standard form. Characteristic
{a) 5,790,000 5.79 * 10° 6
(b) 0.000283 2.83 X 1074 —4
{c) 44 4.4 X 10! 1
{d) 0.623 6.23 X 107 —i
(e) 8.15 8.15 X 10° 0
(f) 461,328 4.61328 X 10° 5
(g) 0.0000005371 5.371 X 107 |
(h) 0.0306 3.06 X 102 ~~
(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-
verse 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 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 X 10'. Hence, 5.79 K 2.83 K 10?
= 1.639 * 10! X 102 = 1.639 « 10° = 1639. If, however, one has
5,790,000 + 0.000283, the use of standard form yields
5.79 X 10°
283 X 10-4 = 204 « 10°" = 2.04 & 10.18
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 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 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 1.65 x 8.9.
Fstimate 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 /eff index is placed under
the hairline, chen 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 BX ¢cXdXe-s:
Set hairline of indicator ac a on D scale.
Move index of C scale under hairline.
Move hairline over 6 on the C scale.
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 numbers have
been set.
Read result under the hairline on the D scale.
PROBLEMS i ANSWERS
lL. 29X34 X 7.5 73.9
2. 17.3 X 43 X 9.2 6,840
3. 343 X 91.5 X 0.00332 167
4+. 19 & 407 X 0.0021 16.24
5. 13.5 X 709 X 0.567 X 0.97 5260
COMBINED MULTIPLICATION AND DIVISION
Many problems call for both multiplication and division,
42 & 37
EXAMPLE: 35
_*See any textbook on elementary algebra. lhe theory of exponents and the rules of operation
a with signed numbers are both involved in a complete treatment of this copic. In this manual it
is assumed that the reader is familiar with this theory.
In scientific work the result would be left in this form, but for popular con- ta =
sumption it would be written as 20,400,000,000. The general rule 1s as follows. A/suve ri Le
**The product of 382 X 165 could now be read under the hairline on the D scale, but this is
® not necessary.
*In using this rule, “first” is to be counted from the left; thus, in 3246, the digit 3 is “first.” : =
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First, set the division of 42 by 65; thar 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 rata is about
equal to i the resule is about two-thirds of 37, or 23.9.
273 & 548
EXAMPLE: 402 X< 34a
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.
; axXcXKexXg::
In general, to do computations of the type bxdXfXh
, sec 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 che numerator ¢, move
the slide to set next factor in denominator, ¢, under the hairline. Continue
moving hairline and slide alternately for other factors (¢, fy g, 4, ekc.). 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 ts under the C index.
Sometimes the slide must be moved so that one index replaces the other. **
2.2 X 2.4
EXAMPLE: sq
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 co the right (so that the /eft index falls under the 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-
nominator 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
mx n ) % aX
PROBLEMS ANSWERS
1. 27 x 48 61.1
19
9. 6.17 X 1.25 XK 9.33 2.07
4.3 X 6.77
3, 842 XK 2.41 X 173 53.7
567 X 11.52
4, 1590 X 3.64 x 0.763 1.081
4.39 < 930
*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 che use of other scales.
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5. 0.0237 < 3970 «K 32 X 6.28
0.00029 X 186000 351
6, 231 X 58.6 X 4930
182.5 X 3770 —_
7 875 X 1414 x 2.01
661 X 35.9 IAS
g. 558 X 1145 x 633 x 809
417 X 757 X 384 2980
9, 0.691 X 34.7 x 0.0561 0.000147
91,500 or 1.47 X 10-5
10. 19.45 X 7.86 X 361 X 64.4
326X974 Le
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 } 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 the numbers
for all pairs of opposite graduttions have the same ratio. Suppose one of the
: : ; . @ ¢
terms of a proportion is unknown. The proportion can be written as a
x
where a, 6, and c, are known numbers and x is to be found,
Rule: Set 2 on the C scale opposite 6 on the D scale. Under ¢ on the C
scale read x on the D scale.
% Lex
3
EXAMPLE: Find x ify =
Set 3 on C opposite 4 on D. Under 5 on C read 6.67 on D.
The proportion above could also be written ~ = —, or “inverted,” and
aoc
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
teads as follows:
Set 2 on the D scale opposite 6 on the C scale. Over ¢ on the D scale
tread 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 proportion written
c
in the form . =4
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b
Proportions can also be solved algebraically. Then ; = < becomes x= _
’
and this may be computed as combined multiplication and division.
PROBLEMS ANSWERS
1 * _ 13.2 300
425 187 ,
2. B05 __ ae 40.7
“x 1.54
3.43.6 x 1247.
89.2 2550
4, 0.063 _ 34.1 276.
0.51 x
5 - 13 65.7
x
SPECIAL GRADUATIONS — CIRCULAR MEASURE
A special graduation on the C scale at r/4=0.7854 is especially convenient
in finding the area or the diameter of a circle, and in similar problems. Since
A= (1/4) d*, where d is the diameter, the following rule applies.
Rule: To find the area (A) of a circle whose diameter d is known, set the
right index of the C scale over d on the D scale. Move the hairline over .7854
mark on righthand section of B scale. Read area under hairline on A scale.
Rule: To find the diameter of a circle whose area A is known, set hairline
over area on A scale (See page 19 for rule regarding section of A scale
to use). Move slide until .7854 mark on righthand section of B scale is under
hairline. Read diameter on D scale below right index of C scale.
A special graduation on the C and D scales at 57.3 is indicated by a small R.
Since one radian=57.3° (approximately), this R graduation is useful in
changing radian measure to degree measure, and conversely.
Rule: ‘To convert radians to degrees, set an index of the C scale to the num-
ber of radians on D. Under R on C read che number of degrees on D. To
convert degrees to radians, set R on C over the number of degrees on D. At
the index of C read the number of radians on D.
EXAMPLES:
(a) Convert 7/2 radians to degrees. Set 2 on C over + on D. At the left-
index of C read 7/2, or 1.57, on D.
(b) Convert 30° to radians. Set R on C over 30 on D. At the index of C
read 0.524 radians on D.
Note that this method puts the result on the D scale in convenient form
for continued computation. In similar fashion, by interchanging C and D and
the words “under” and “over in the above rules, the result can be put on the
C scale.
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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 1s a C scale which increases from right to left. It
may be used for finding reciprocals. When any number is set under the hair-
line on the C scale its reciprocal is found under the hairline on the CI scale,
and conversely.
EXAMPLES:
(a) Find 1/2.4. Set 24.0n 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.
The CI scale is useful in replacing a division by a multiplication. Since
a
f= 4X 1/6, 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
tule.
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 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 4s 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 XxX (1/2.79).
4.13 & 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
readily, Also, it is sometimes easier to Bet a X b by setting the hairline on a,
pulling 4 on the CI scale under the 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 reciprocals.
The DI scale has several important uses, of which rhe following is representa-
tive.
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i 1 licated expression
Expressions of the type 1/X, where X is some comp
or formula may be computed by first finding the value of X. If the result
for X falls ai D, then 1/X may be read under the hairline on DI.
EXAMPLE:
Gre wv 12a Pp y ] S es.
nd ult } 0 OX 38 usin the C and :B) cal
Read the reciprocal .0273 under the hairline on the DI scale. Or set the hairjine
5 : der the hairline, and read the
265 of the DI scale, pull 138 of the C scale un Jin | rea
result on the D scale under the left index of the C scale. This is equivalent to
. (1/.265)
writing the expression as ~Tag7 -
PROBLEMS ANSWERS
1. 1 143
7
2, 1 .0284
35.2
—_— 5.57
1795
4, 1 0001555
6430
5. 1 318
7
6 1 240
00417
THE CF/z AND DF/z SCALES
It should be understood that the use of che CF and DF 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
certain settings. In this way the speed can be increased and errors minimized.
When = on the C scale is opposite the right index of the D scale, about half
the slide projects beyond the rule. If this part were cut off and used to fill
in the opening at the left end, the result would be a “‘folded”’ C scale, or CF
scale. Such a scale is printed at the top of the slide. It begins at x and the
index is near the middle of the rule. The DF scale is similarly placed. Any
setting of C on D is automatically set on CF and DF. Thus if 3 on C is
opposite 2 on D, then 3 on CF is also opposite 2 on DF. The CF and DF
scales can be used for multiplication and division in exactly the same way as
the C and D scales.
The most important use of the CF and DF scales is to avoid resetting the
slide. Ifa setting of the indicator cannot be made on the C or D scale, it can
be made on the CF or DF scale.
EXAMPLES:
(a) Find 19.2 < 6.38. 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
scale, and read the result 122.5 on the DF scale. Or set the index of CF on
19.2 of DF. Move indicator to 6.38 on CF and read 122.5 on DF.
(b) Find S20 5868 Set 5.72 on C opposite 8.39 on D. The indicator
cannot be moved to 9.65 of C, but it can be moved to this setting on CF and
the result, 14.15, read on DF. Or the entire calculation may be done on the
CF and DF scales.
These scales are also helpful in calculations involving x and 1/m. When the
indicator is set on any number N on D, the reading on DF is Nx. This can
be symbolized as (DF) = 7(D). Then (D) = ee This leads to the fol-
: T
lowing simple rule.
Rule: If the diameter of a circle is set on D, the circumference may be
read immediately on DF, and conversely.
EXAMPLES:
(a) Find 5.6%. Set indicator over 5.6 on D. Read 17.6 under hairline on DF.
(b) Find 8/7. Set indicator over 8 on DF. Read 2.55 under hairline on D.
— (c) Find the circumference of a circle whose diameter is 7.2. Set indicator on
/ 7.2 of D, Read 22.6 on DF.
L/si DE RULE
pr (d) Find the diameter of a circle whose circumference is 121. Set indicator
on 121 of DF. Read 38.5 on D.
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Finally, these scales are useful in changing radians to degrees and con-
versely. Since 7 radians = 180 degrees, the relationship may be written as a
a. , © ra
proportion 7" igo r= = 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 conventent settings as suggested by the proportion.
Thus one can set the ratio 7/180 on the CF and DF scales and find the 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.
PROBLEMS ANSWERS
1. 1.414 X 7.79 11.02
2. 2.14 X 57.6 123.3
3, 84.5 X 7.59 17.43
36.8 :
4. 265 X 4 8.33
5. .1955 X 23.7
50.7 X r AEE,
THE CIF SCALE
Like the other special scales the CIF scale does not increase the power of
the instrument to solve problems. Ic 1s 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 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 co 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 X 8.4 X 16.1. Set indicator on 2.07 on C. Move slide until
8.4 on Cl is under hairline. Move hairline ro 16.1 on C. Read 280 on D under
hairline. Or, set the index of