The Polyphase Duplex Decitrig Slide Rule No. 4071: A Manual (1939 edition, with the later Manual No. 68 2055 edition covers)
K&E
REG. U. S. PAT. OFF.
POLYPHASE DUPLEX
DECITRIG
TRADE MARK
Slide Rule
No. 4071
A MANUAL
—
BY
LYMAN M. KELLS, PH.D.
Associate Professor of Mathematics
WILLIS F. KERN
Assistant Professor of Mathematics
AND
JAMES R. BLAND
Assistant Professor of Mathematics
Allat the United Siates Naval Academy
PUBLISHED By
KEUFFEL & ESSER CO.
NEW YORK. 127 Fulton Street, General Office and Factories, HOBOKEN, N. J.
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Drawing Materials, Mathematical and Surveying Instruments, Measuring Tapes.
K&E
REG. U. S. PAT. OFF.
POLYPHASE DUPLEX
DECITRIG
TRADE MARK
Slide Rule
No. 4071
A MANUAL
BY
LYMAN M. KELLS, PH.D.
Associate Professor of Mathematics
WILLIS F. KERN
Assistant Professor of Mathematics
AND
JAMES R. BLAND
Assistant Professor of Mathematics
All at the United States Naval Academy
PUBLISHED By
KEUFFEL & ESSER CO.
NEW YORK, 127 Fulton Street, General Office and Factories, HOBOKEN. N. J.
CHICAGO ST. LOUIS DETROIT
516-20 S. Dearborn St. 817 Locust St. General Motors Building.
SAN FRANCISCO LOS ANGELES MONTREAL
30-34 Second St. 730 S. Flower St. 7-9 Wotre Dame St., W.
Drawing Materials, Mathematical and Surveying Instruments, Measuring Tapes.
COPYRIGHT, 1939,
KEUFFEL & ESSER CO.
PRINTED IN U.S.A.
SLIDE RULE
MANUAL
POLYPHASE
DUPLEX DECITRIG
By
LYMAN M. KELLS,
WILLIS F. KERN
and
JAMES R. BLAND
MANUAL
No. 68 2055
OLD NO. 4187U
KEUFFEL & ESSER CO.
ie ae te RSS
Kee
POLYPHASE DUPLEX
TRADE MARK ®
DECITRIG
SLIDE RULE
MANUAL
BY
LYMAN M. KELLS, PH.D.
Professor Emeritus of Mathematics
WILLIS F. KERN
Formerly Associate Professor of Mathematics
AND
JAMES R. BLAND
Professor of Mathematics
All at the United States Naval Academy
PUBLISHED BY
KEUFFEL & ESSER CoO.
NEW YORK + HOBOKEN, N. J.
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© 1939, 1962
KEUFFEL & ESSER CO.
All Rights Reserved.
PRINTED IN U.S.A.
12992
THE K& E POLYPHASE DUPLEX DECITRIG SLIDE RULE
No. 4071
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PREFACE
This slide rule manual has been written for study without
the aid of a teacher. For this reason one might suspect that
the treatment is superficial. On the contrary, however, the
subject matter is so presented that the beginner uses two general
principles while he is learning to read the scales and perform the
simpler operations. The mastery of these two principles gives
the power to devise the best settings for any particular purpose,
and to recall settings which have been forgotten.
These principles are so simple and so carefully explained
and illustrated both by diagram and by example that they are
easily mastered. In Chapter II, they are applied to simple preb-
lems in multiplication and division; in Chapters III, IV, V
they are used to solve problems involving multiplication, division,
square and cube root, trigonometry, logarithms, and powers of
numbers. Chapter V_ explains the slide rule from the logarithmic
standpoint. Those who desire a theoretical treatment are likely
to be surprised to find that the principles of the slide rule are so
easily understood in terms of logarithms.
CONTENTS
CHAPTER 1
MULTIPLICATION AND DIVISION
Arr. Pacs
L.cIntroduction « ss ¢ % 4 4g; BES Se a KS SEER & . 8
2. Reading thescales 2... 56 tt tt ttt 3
3. Accuracy of the sliderule . 2... 2... 2 ee ee ee 6
4. Definitions. .. 2... 2... ee ee “seems ee
5. Multiplication . 0... 6 2 ee ee ew eee ees 7
6., Hither index may be'used . 2... 66 ie hs we ee ee we 9
Ce DAVBOD ice: oe se wRRE K S te oe ESE w w w em 9
8. Simple applications, percentrge, rates. 2... 2... ee ee 10
9. Use of the scales DF and CF (folded seales) . . . ......... 12
CHAPTER II
THE PROPORTION PRINCIPLE AND COMBINED OPERATIONS
10.
ll.
12.
13.
14.
15.
16,
Introduction © 5 6.8 © - eum n be pa ck MERE R RES eee 14
Use of proportions . 2. 2... ee 14
Forming proportions from equations... . 2... 2. ee ee 18
Equivalent expressions of quantity . . 2... 2... 2 eee . 19
The CJ and CIF (reciprocal) seales . 2 2. 2 2. 1. ee ee ee 20
Proportions involving the C7 scale . 2... 2... ee 22
Combined multiplication and division. . 2. 2... ee 23
CHAPTER IIT
SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS
AZ OQUAROM S75) HS S es BG we eee ee ee SL EO cess 26
18. Square roots. 2 2. 2 27
19. Evaluation of simple expressions containing square roots and squares . 28
20. Combined operations involving square roots and squares... . . . 30
21, Cube, 2 aaa KG BU OG WE RE ww aS we me ew OF 32
22; Cube TOG. wes eo 6 wos wo Ee Ge we GOR Rw EE 33
23. Combined operations ......... ee WF SG Hw & BNE Fe 34
24. The Lecale . 2... sss ele Ek ee eo ie ee 6 cic as 386
CHAPTER IV
PLANE TRIGONOMETRY
ART. Pace
25: Some important formulas from plane trigonometry ........ . 37
26. The S (Sine) and ST (Sine Tangent) scales . ........... 38
oT: The T (Tangent)'e¢ale gc a ORO OS OS Ow Oe GE ew HUEY 8 40
28. The red numbers on the Sand 7T'scales ...........---. 40
20) Combiisd Gperatone: goin sp we et ee EO OMe et ROS WE 42
30. The use of the trigonometric scales in solving proportions ..... . 44
31. Law of sines applied to right triangles . 2... 2... 2... 1 wwe 46
32. The law of sines applied to right triangles with two iegs given ... . 50
33. Law of sines applied to oblique triangles when two opposite parts are
ROWE: cms kK ES Be eS ow Rw GE A ee eS 51
34. Law of sines applied to oblique triangles, continued. Ambiguous case . 53
35. Law of sines applied to oblique triangles in which two sides and the
included angle‘aregivenss: 2 GS SRE KRIS RE RST ER Sw: 55
36. Law of cosines applied to oblique triangles in which three sides are given 56
STASUMMATY 4 aH EG GK ON Sw OR ee Sw ee we aOR 58
38. Miscellaneous Exercises . . 2... 1 1 ee ee ee ee 59
39. To change radians to degrees or degrees toradians ......... 59
40. Sines and tangents of smallangles .............2.4.-., 60
41. Applications involving vectors . . 2. 1 6 ew ee ee ee ee 62
£3: .-Applications cs cis wie eS RK HEL MS Rw I Sw wo 66
CHAPTER V
LOGARITHMS AND THE SLIDE RULE
43. Construction of the Dscale ...........-2-228++26 70
af; Accuracy, 6 5 i GG 6 RR Se ee RS Pe A RS ee a 71
45. Multiplication and division . 2... 2... eee ee ee eee 71
46; "Theanverted:aeales: c= «6 ss HG ED Ow me GE Sw mE ww 72
47. The A scale, the B scale, and the K scale... .......---. 74
48. The trigonometric scales. . 2. 2 1 1 1 1 ee ee ee et tt ee 74
49. Twoapplicationgc05: 6 cs howe TR ee eS eS 77
Historical note. . 2. 2. ee ee 78
AMSWEFS cers ee GONSE wi BONE & Sw NSE Bw Ww KEES Se KU! Ew 80
Indexs % 4 & 8 6 GS SS HR o 2 % Sei & ES a ES GES . 86
CHAPTER T
MULTIPLICATION AND DIVISION
1. Introduction. This pamphlet is designed to enable any in-
terested person to learn to use the slide rule efficiently. The beginner
should keep his slide rule before him while reading the pamphlet,
should make all settings indicated in the illustrative examples, and
should compute answers for a large number of the exercises. The
principles involved are easily understood but a certain amount of
practice is required to enable one to use the slide rule efficiently and
with a minimum of error.
2. Reading the scales.* Everyone has read a ruler in measuring
a length. The number of inches is shown by a number appearing
on the ruler, then small divisions are counted to get the number of
16th’s of an inch in the fractional part of the inch, and finally in
close measurement, a fraction of a 16th of an inch may be estimated.
We first read a primary length, then a secondary length, and finally
estimate a tertiary length. Exactly the same method is used in
reading the slide rule. The divisions on the slide rule are not
uniform in length, but the same principle applies.
Figure 1 represents, in skeleton form, the fundamental scale of
the slide rule, namely the D scale. An examination of this actual
oj re ee eae
Fig. 1.
scale on the slide rule will show that it is divided into 9 parts
by primary marks which are numbered 1, 2,3,...,9, 1. The space
between any two primary marks is divided into ten parts by nine
secondary marks. These are not numbered on the actual scale except
*The description here given has reference to the 10” slide rule. However
anyone haying a rule of different length will be able to understand his rule in
the light of the explanation given.
3
4 MULTIPLICATION AND DIVISION {Cuap. I
between the primary marks numbered 1 and 2. Fig. 2 shows the
secondary marks lying between the primary marks of the D scale. On
this scale each italicized number gives the reading to be associated
dD | | ! if rir WRAL TAM TTTT]TT ETE |
Fia. 2,
with its corresponding secondary mark. Thus, the first secondary
mark after 2 is numbered 21, the second 22, the third 23, ete.; the
first secondary mark after 3 is numbered 31, the second 32, ete.
Between the primary marks numbered | and 2 the secondary marks
are numbered 1, 2,...,9. Evidently the readings associated with
these marks are 11, 12, 13,...,19. Finally between the secondary
marks, see Fig. 3, appear smaller or tertiary marks which aid in
D Fa 3 Raae 6
Scole 0
Fic. 3.
obtaining the third digit of a reading. Thus between the secondary
marks numbered 22 and 23 there are 4 tertiary marks. If we think of
the end marks as representing 220 and 230, the four tertiary marks
divide the interval into five parts each representing 2 units. Hence
with these marks we associate the numbers 222, 224, 226, and 228;
similarly the tertiary marks between the secondary marks numbered
32 and 33 are read 322, 324, 326, and 328, and the tertiary marks
between the primary marks numbered 3 and the first succeeding
secondary mark are read 302, 304, 306, and 308. Between any pair
of secondary marks to the right of the primary mark numbered 4,
there is only one tertiary mark. Hence, each smallest space repre-
sents five units. Thus the tertiary mark between the secondary
marks representing 41 and 42 is read 415, that between the secondary
marks representing 55 and 56 is read 555, and the first tertiary
mark to the right of the primary mark numbered 4 is read 405.
The reading of any position between a pair of successive tertiary
marks must be based on an estimate. Thus a position half way
between the tertiary marks associated with 222 and 224 is read 223
and a position two fifths of the way from the tertiary mark numbered
415 to the next mark is read 417. The principle illustrated by these
readings applies in all cases.
§2] READING THE SCALES 5
Consider the process of finding on the D scale the position rep-
resenting 246. The first figure on the left, namely 2, tells us that
the position lies between the primary marks numbered 2 and 3.
This region is indicated by the brace in figure (a). The second
figure from the left, namely 4, tells us that the position lies
~
C] 2 3 ty)
of 5 } 1
Fic. a.
between the secondary marks associated with 24 and 25. This region
is indicated by the brace in Fig. (b). Now there are 4 marks be-
jt el Meer ere, pi tiyi il
pi hh kk bh bs 6 lb bby! PITT TiiTy
Fic. b.
Lipitor ty
rrrrytriryt
tween the secondary marks associated with 24 and 25. With these
are associated the numbers 242, 244, 246, and 248 respectively.
olen a
c 3 ”
t) pee Sioa aay yg” ty ?
Fic. c.
Thus the position representing 246 is indicated by the arrow in
Fig. (c). Fig. (abe) gives a condensed summary of the process.
246 LIES BETWEEN 240 AND 250
7 8 ; 9 ‘ rn
7 8 9 7
Fic. abe.
{t is important to note that the decimal point has no bearing upon
the position associated with a number on the C and D scales. Con-
sequently, the arrow in Fig. (abc) may represent 246, 2.46, 0.000246,
24,600, or any other number whose principal digits are 2,4, 6. The
placing of the decimal point will be explained later in this chapter.
For a position between the primary marks numbered 1 and 2, four
digits should be read; the first three will be exact and the last one
6 MULTIPLICATION AND DIVISION [Cuap. I
estimated. No attempt should be made to read more than three
digits for positions to the right of the primary mark numbered 4.
While making a reading, the learner should have definitely in mind
the number associated with the smallest space under consideration.
Thus between 1 and 2, the smallest division is associated with 10 in
the fourth place; between 2 and 3, the smallest division has a value
2 in the third place; while to the right of 4, the smallest division
has a value 5 in the third place.
The learner should read from Fig. 4 the numbers associated with
the marks lettered A, B, C, ... and compare his readings with the
Fie 4.
following numbers: A 365, B 327, C 263, D 1745, E 1347, F 305,
G 207, H 1078, 1435, J 427.
3. Accuracy of the slide rule. From the discussion of §2, it
appears that we read four figures of a result on one part of the
scale and three figures on the remaining part. This means an attain-
able accuracy of roughly 1 part in 1000 or one tenth of one per cent.
The accuracy is nearly proportional to the length of the scale.
Hence we associate with the 20 inch seale an accuracy of about one
part in 2000, and with the Thacher Cylindrical slide rule, an accu-
racy of about one part in 10,000. The accuracy obtainable with
the 10 inch slide rule is sufficient for most practical purposes; in
any case the slide rule result serves as a check.
4. Definitions. The central sliding part of the rule is called
the slide, the other part the body. The glass runner is called the
indicator and the line on the indicator is referred to as the hairline.
The mark associated with the primary number 1 on any scale is
called the index of the scale. An examination of the D scale shows
that it has two indices, one at the left end and the other at the right
end.
Two positions on different scales are said to be opposite if, without
moving the slide, the hairline may be brought to cover both positions
at the same time.
§5] MULTIPLICATION 7
5. Multiplication.
The process of multiplication may be per-
formed by using scales C and D. The C scale is on the slide, but in
other respects it is like the D scale and is read in the same manner
To multiply 2 by 4,
to 2 on D set index of C,
push hairline to 4 on C,
at the hairline read 8 on D.
I PF
{_ t a }
Fic. 5 (a).
SA, VE |
or
2 aw J 0 wae . 2. . 1 2 i
o[s 2 xd 3 ' x ea th 7 7 Terres
Ves PR : t ‘ + L t i r)
Sa EE
Fic. 5 (6).
Fig. 5(b) shows the rule set for multiplying 2 by 4 and Fig. 5(a)
shows the same setting in skeleton form. To multiply 3 x 3
to 3 on D set index of C,
push hairline to 3 on C,
at the hairline read 9 on D.
See Fig. 6(a) for the setting in skeleton form and Fig. 6(b) for a
photograph of the setting.
°
oe
ot—u
an}
j
Fic. 6 (a).
Fic. 6 (6).
8 MULTIPLICATION AND DIVISION [Cuap. I
To multiply 1.5 x 3.5, disregard the decimal point and
to 15 on D set index of C,
push hairline to 35 on C,
at the hairline read 525 on D.
By inspection we know that the answer is near to 5 and is there-
fore 5.25.
To find the value of 16.75 x 2.83 (see Fig. 7(a) and Fig. 7(0) )
283
| ¢} ieccctat tf t/
fod LELULELE ULE 4 ally ae oe a a
1675
474
Fic. 7 (a).
4
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ci Kaa ata sstltostaRatcttcbtctobn aon tatatataLetetott MitetetstaletetototePetatetetalatetabet isnt
Cc} B Pp. ea es WF ER ed MOM RNR ag Bi
ry rm aaa | dd kA ld ie ii iia A
[as ENE LPSRI OT OSE LOE Oe See Pay Owe
Fic. 7 (6).
disregard the decimal point and
to 1675 on D set index of C,
push hairline to 283 on C,
at the hairline read 474 on D.
To place the decimal point we approximate the answer by noting
that itisnearto3 x 16= 48. Ilence the answer is 47.4.
These examples illustrate the use of the following rule.
Rule. To find the product of two numbers, disregard the decimal
points, opposite either of the numbers on the D scale set the index of
the C scale, push the hairline of the indicator to the second number
on the C scale, and read the answer under the hairline on the D scale.
The decimal point is placed in accordance with the result of a rough
calculotion.
EXERCISES
1.32. 8. 2.03 x 167.3.
2. 3.5 x 2. 9. 1.536 x 20.6.
3.5 x 2. 10. 0.0756 x 1.093.
4. 2 x 4.55. 11. 1.047 x 3080.
5. 4.5 x 1.5. 12. 0.00205 x 408.
6. 1.75 x 5.5. 13. (3.142)?
7. 4.33 x 11.5. 14. (1.756)2
§7] DIVISION 9
6. Either index may be used. It may happen that a product
cannot be read when the left index of the C scale is used in the rule
of §5. This will be due to the fact that the second number of
the product is on the part of the slide projecting beyond the body.
In this case reset the slide using the right index of the C scale in
place of the left, or use the following rule: when a number is to be
read on the D scale opposite a number of the C scale and cannot be
read, push the hairline to the index of the C scale inside the body and
draw the other index of the C scale under the hairline.* The desired
reading can then be made.
If, to find the product of 2 and 6, we set the left index of the
C scale opposite 2 on the D scale, we cannot read the answer on
the D scale opposite 6 on the C scale. Hence, we set the right index
of C opposite 2 on D; opposite 6 on C read the answer, 12, on D.
Again, to find 0.0314 x 564,
to 314 on D set the right index of C,
push hairline to 564 on C,
at the hairline read 1771 on D.
A rough approximation is obtained by finding 0.03 x 600 = 18.
Hence the product is 17.71.
EXERCISES
Perform the indicated multiplications.
1.3 x 5. 5. 0.0495 x 0.0267.
2. 3.05 x 5.17. 6. 1.876 x 926.
3. 5.56 x 634. 7. 1.876 x 5.32.
4. 743 x 0.0567. 8. 42.3 x 31.7.
7. Division. The process of division is performed by using the
C and D scales.
To divide 8 by 4 (see Figs. 8(a) and 8(6) ),
push hairline to 8 on D,
draw 4 of C under the hairline,
opposite index of C read 2 on D.
[ey
G4
ot
Laos
Fic. 8 (a).
*This rule, slightly modified to apply to the scales being used, is generally
applicable when an operation calls for setting the hairline to a position on the
part of the slide extending beyond the body.
To divide 876 by 20.4,
push hairline to 876 on D,
draw 204 of C under the hairline,
opposite index of C read 429 on D.
The rough calculation 800 + 20 = 40 shows that the decimal point
must be placed after the 2. Hence the answer is 42.9.
These examples illustrate the use of the following rule.
Rule. 1'o find the quotient of two numbers, disregard the decimal
points, opposite the numerator on the D scale set the denominator on
the C scale, opposite the index of the C scale read the quotient on the D
scale. The position of the decimal point is determined from informa-
tion gained by making a rough calculation.
EXERCISES
Perform the indicated operations.
1, 87.5 + 37.7. 6. 2875 + 37.1.
2. 3.75 + 0.0227. 7. 871 + 0.468.
3. 0.685 + 8.93. 8. 0.0385 + 0.001462.
4. 1029 + 9.70. 9. 3.14 + 2.72.
5. 0.00377 + 5.29. 10. 3.42 + 81.7.
8. Simple applications, percentage, rates. Many problems in-
volving percentage and rates are easily solved by means of the slide
rule.
One per cent (1%) of anumber N is N x 1/100; hence 5% of N is
N x 5/100, and, in general, p% of Nis pN/100. Hence to find 83%
of 1872
to 1872 on D set right index of C,
push hairline to 83 on C,
at the hairline read 1554 on D.
Since (83/100) x 1872 is approximately a x 2000 = 1600, the
answer is 1554.
§3) SIMPLE APPLICATIONS 11
To find the answer to the question ““M is what per cent of N?”’ we
must find 100 M +N. Thus, to find the answer to the question
“87 is what per cent of 184.7?” we must divide 87 x 100 = 8700
by 184.7. Hence
push hairline to 87 on D,
draw 1847 of C under the hairline,
opposite index of C read 471 on D.
= 45 shows that the decimal point should
The rough calculation 00
be placed after the 7. Hence the answer is 47.1%.
For a body moving with a constant velocity, distance = rate times
time. Hence if we write d for distance, r for rate, and ¢ for time, we
have
d=rt,orr=-tort= é,
t r
To find the distance traveled by a car going 33.7 miles per hour
for 7.75 hours, write d = 33.7 x 7.75, and
to 337 on D set right index of C,
push hairline to 775 on C,
at hairline read 261 on D.
Since the answer is near to 8 x 30 = 240 miles, we have d = 261 miles.
To find the average rate at which a driver must travel to cover
287 miles in 8.75 hours, writer = 287 + 8.75, and
push hairline to 287 on D,
draw 875 of C under the hairline,
opposite the index of C read 328 on D.
Since the rate is near 280 + 10 = 28, we have r = 32.8 miles per hour
EXERCISES
1. Find (a) 86.3 per cent of 1826.
(b) 75.2 per cent of 3.46.
(c) 18.3 per cent of 28.7.
(d) 0.95 per cent of 483.
2. What per cent of
(a) 69 is 18?
(b) 132 is 85?
(c) 87.6 is 192.8?
(d) 1027 is 28?
3. Find the distance covered by a body moving
(a) 23.7 miles per hour for 7.55 hours.
(b) 68.3 miles per hour for 1.773 hours.
(c) 128.7 miles per hour for 16.65 hours.
12 MULTIPLICATION AND DIVISION [Cuar. J
4. At what rate must a body move to cover
(a) 100 yards in 10.85 seconds.
(b) 386 feet in 25.7 seconds.
(c) 93,000,000 miles in 8 minutes and 20 seconds.
5. Find the time required to move
(a) 100 yards at 9.87 yards per second.
(b) 3800 miles at 128.7 miles per hour.
(c) 25,000 miles at 77.5 miles per hour.
9. Use of the scales DF and CF (folded scales). The DF and
the CF scales are the same as the C and the D scales respectively
except in the position of their indices. The fundamental fact con-
cerning the folded scales may be stated as follows: if for any setting
of the slide, a number M of the C scale is opposite a number N on
the D scale, then the number M of the CF scale is opposite the number N
on the DF scale. Thus, if the learner will draw 1 of the CF scale
opposite 1.5 on the DF scale, he will find the following opposites on
the CF and DF scales
and the same opposites will appear on the C and D scales.
In accordance with the principle stated above, if the operator
wishes to read a number on the D scale opposite a number N on the
C scale but cannot do so, he can generally read the required number
on the DF scale opposite N on the CF scale. For example to find
2 x 6,
to 2 on D set ieft index of C,
push hairline to 6 on CF,
at the hairline read 12 on DF.
By using the CF and DF scales we saved the trouble of moving the
slide as well as the attendant source of error. This saving, entering
as it does in many ways, is a main reason for using the folded scales.
The folded scales may be used to perform multiplications and divi-
sions just as the C and D scales are used. Thus to find 6.17 x 7.34,
to 617 on DF set index of CF,
push hairline to 734 on CF,
at the hairline read 45.3 on DF;
§91 USE OF SCALES DF AND CF 18
or
to 617 on DF set index of CF,
push hairline to 734 on C,
at the hairline read 45.3 on D.
Again to find the quotient 7.68 /8.43,
push hairline to 768 on DF,
draw 843 of CF under the hairline,
opposite the index of CF read 0.912 on DF;
or
push hairline to 768 on DF,
draw 843 of CF under the hairline,
opposite the index of C read 0.912 on D.
It now appears that we may perform a multiplication or a division
in several ways by using two or more of the seales C, D, CF, and DF.
The sentence written in italics near the beginning of the article sets
forth the guiding principle. A convenient method of multiplying or
dividing a number by <= ( = 3.14 approx.) is based on the statement:
any number on DF is = times its opposite on D, and any number on
Dis 1/= times its opposite on DF.
EXERCISES
Perform each of the operations indicated in exercises 1 to 11 in four ways;
first by using the C and D scales only; second by using the CF and DF scales only;
third by using the C and D scales for the initial setting and the CF and DF scales
for completing the solution; fourth by using the CF and DF scales for the initial
setting and the C and D scales for completing the solution.
1. 5.78 x 6.35. 9. 0.0948 + 7.23.
2. 7.84 x 1.065. 10. 149.0 + 63.3.
3. 0.00465 + 73.6. 11. 2.718 + 65.7.
4. 0.0634 x 53,600. 12. 783%.
5. 1.769 + 496. 13. 783 +.
6. 946 + 9.0677. 14. 0.0876 =z.
7. 813 x 1.951. 15. 0.504 +7.
8. 0.00755 + 0.338. 16. 1.072 + 10.97:
CHAPTER II
THE PROPORTION PRINCIPLE AND
COMBINED OPERATIONS
10. Introduction. The ratio of two numbers a and b is the quotient
of a divided by b or a/b. A statement of equality between two
ratios is called a proportion. Thus
2 62 7 aie
3°9'5 11’ b d
are proportions. We shall at times refer to equations having such
forms as
10
2
wilbo
=
5
eico
ac e
yaad 5 = 3" 7
as proportions.
An important setting like the one for multiplication, the one for
division, and any other one that the operator will use frequently should
be practiced until it ts made without thought. But, in the process of
devising the best settings to obtain a particular result, of making a
setting used infrequently, or of recalling a forgotten setting, the applica-
tion of proportions as explained in the next article is very useful.
11. Use of Proportions. If the slide is drawn to any position,
the ratio of any number on the D scale to its opposite on the C scale
is, in accordance with the setting for division, equal to the number
on the D scale opposite the index on the C scale. In other words,
when the slide is set in any position, the ratio of any number on the
D scale to its opposite on the C scale is the same as the ratio of any
other number on the D scale to its opposite on the C scale. For example
25
fo 3 5 48 si |
5
Fic. 1.
14
§11] USE OF PROPORTIONS 15
draw 1 of C opposite 2 on D (see Fig. 1) and find the opposites
indicated in the following table:
corcr)| 1 | 15 | 25 | » | 4 | 5 |
D(orDF} 2 | 3 | 5 | 6 | 8 | 10 |
and draw 2 of C over 1 on D and read the same opposites. The same
statement is trze if in it we replace C scale by CF scale and D scale
by DF scale. Hence, #f both numerator n and denominator d of a
ratio in a given proportion are known, we can set n of the C scale opposite
d on the D scale and then read, for an equal ratio having one part
known, its unknown part opposite the known part. We could also
begin by setting d on the C scale opposite n on the D scale. It
is important to observe that all the numerators of a series of equal
ratios must appear on one scale and the denominators on the other.
For example, let it be required to find the value of z satisfying
x 9
5677? (1)
Here the known ratio is 9/7. Hence
push hairline to 7 on D,
draw 9 of C under the hairline,
push hairline to 56 on D,
at the hairline read 72 on C.
or
push hairline to 9 on D,
draw 7 of C under the hairline,
push hairline to 56 on C,
at the hairline read 72 on D.
The CF and DF scales could have been used to obtain exactly the
same settings and results.
For convenience we shall indicate the settings for solving (1) as
follows:
ds Lisritisif [ f {
o ¢ 5 | | 4 }
Fic. 2.
16 THE PROPORTION PRINCIPLE (Cuap. II
C (or CF) set 7 opposite 56 (2)
nas 2
D (or DF) to 9 read x ( = 72)
C (or CF) set 9 read x ( = 72) (3)
3
D (or DF) to 7 | opposite 56
The letter at the beginning of a line indicates that all numbers
on that line are to appear on the scale designated by that letter,
and any pair of numbers set opposite each other in the frame are to
appear as opposites on the slide rule.
To find the values of x, y, and z defined by the equations
Cc. 3.15 2x 576 2
D* 5.29 4.35 y 183.4’
we observe that 3.15 /5.29 is the known ratio, and
push hairline to 529 on D,
draw 315 of C under the hairline;
opposite 435 on D, read x = 2.59 on C;
opposite 576 on C, read y = 96.7 on D;
opposite 1834 on D, read z = 109.2 on C.
The positions of the decimal points were determined by noticing
that each denominator had to be somewhat less than twice its asso-
ciated numerator because 5.29 is somewhat less than twice 3.15. The
setting may appear simpler when written in the following form
analogous to (2):
C |\set 3.15jread x ( = 2.59)| oppos‘te 57.6 |read z (= 109.2)
aa (4)
D || to 5.29 opposite 183.4
opposite 4.35 read y (= 96.7
When an answer cannot be read, apply the italicized rule of §6.
Thus to find the values of x and y satisfying
§11] USE OF PROPORTIONS Li
to 97t;'on D set 1456 of C; then, since the answers cannot be read,
push the hairline to the index on C, draw the right index of C under
the hairline and
opposite 587 on D, read x = 87.6 on C;
opposite 578 on C, read y = 38.7 on D.
Here the positions of the decimal points were determined by observing
that each denominator had to be about six times the associated
numerator.
When a result cannot be read on the C scale nor on the D scale
it may be possible to read it on the CF scale or on the DF scale.
Thus, to find z and y satisfying the equations
C(orCF) . 4.92 1 yy
D (or DF) * z 3.23 13.08’
to 323 on D set left index of C;
opposite 492 on CF, read x = 15.89 on DF;
opposite 1308 on DF, read y = 4.05 on CF.
A slight inspection of the scales will show the value of the state-
ment: If the difference of the first digits of the two numbers of the
known ratio is small use the C and D scales for the initial setting;
if the difference is large use the CF and DF scales. Since in the next
to the last example, the difference between the first digits was great,
the CF and DF scales should have been used for the initial setting.
This would have eliminated the necessity for shifting the slide.
EXERCISES
Find, in each of the following equations, the values of the unknowns.
1 2 =
Pe ae See! abi,
“1.804 25 0.785
3. 2. 246 _ 28
“709° y 384
4. 2 9 _ 8:28, 2.01
* 0.204 0.0506 2 0.1034
5 cs 3 2
°2.07° 61.3 1.571"
6, 8-51 9 _ 235
" 5 “Sg”
7, 17 _ 1.365 _ 4.86
z 8.53 y
18 THE PROPORTION PRINCIPLE [Cuap. II
15
7.34 ~ 29.7
z y 1.076
% BO E- pT 0.287"
»
S18
1
eh
‘ce
12. Forming proportions from equations. Since proportions are
algebraic equations, they may be rearranged in accordance with the
laws of algebra. For example, if
ab
= rs , (1)
we may write the proportion
x ab
, (3)
(4)
Rule (A). A number may be divided by 1 to form a ratio. This
was done in obtaining proportion (2).
Rule (B). A factor of the numerator of either ratio of a proportion
may be replaced by 1 and written as a factor of the denominator of the
other ratio, and a factor of the denominator of either ratio may be replaced
by 1 and written as a factor of the numerator of the other ratio. Thus
(3) could have been obtained from (1) by transferring a from the
numerator of the right hand ratio to the denominator of the left
hand ratio.
For example, to find x , write 2 = ox 28 , apply (B) tu
35 35
Cc. x 28
obtain = Di 16735”
and push hairline to 35 on D,
draw 28 of C under the hairline;
opposite 16 on D, read xz = 12.8 on C.
§13] EQUIVALENT EXPRESSIONS OF QUANTITY 19
To recall the rule for dividing a given number M by a second given
: M a. wD zx M
number N, write x = WV apply rule (A) to obtain G: iv Ww
and push hairline to M on D,
draw N of C under the hairline;
opposite index of C, read x on D.
To recall the rule for multiplication, set x = x , apply rule (B)
a
ai, D X N
to obtain rk ” mas Wey
and to N on D set index of C;
opposite M on C, read x on D.
864 7.48 864
To find xz it > = (7.48) (25.5) , use rule (B) to get —— = 25.5’
make the comeaponding setting and read x = 0.221. The position of
the decimal point was determined by observing that x must be about
1
40 of 8, or 0.2.
EXERCISES
Find in each case the value of the unknown quantity.
86 x 70.8 -, 3.95 x 0.707
1. ¥-——95 6. 0.874 - ——— 3
147.5 x 8.76
2. = 3960 * 7. 2580y = 17.9 x 587.
0.797 x 5.96 0.0879
4, wee, 9, 1 _ 0.772
y * 386 2.85y
$, apes See,
0.563" 10. 3.14y = 0.785 x 38.7.
13. Equivalent expressions of quantity.* When the value of a
quantity is known in terms of one unit, it is a simple matter to
find its value in terms of a second unit. Thus to find the number
of square feet in 3210 sq. in., write
_1__ no. of sq.ft.
144 no. of sq.in. 3210’
*Tables of equivalents may be found in Engineer’s Manuals and in other places.
20 THE PROPORTION PRINCIPLE [Cuap. II
since there are 144 sq. in. in a square foot; hence
to 144 on D, set index of C;
opposite 3210 on D, read x = 22.3 on C;
that is, there are 22.3 sq. ft. in 3210 sq. in.
Again consider the problem of finding the number of nautical
miles in 28.5 ordinary miles. Since there are 5280 ft. in an ordinary
mile and 6080 ft. in a nautical mile, write
5280 _ no. of naut. mi. x
6080 no. of ord. mi. 28.5”
make the corresponding setting and read xz = 24.7 naut. mi.
EXERCISES
1. An inch is equivalent to 2.54 cm. Find the length in cm. of a rod 66 in.
long.
2. The number of meters in a given length is to the number of yards as 171
is to 187. Find the number of meters in a 300 yd. distance.
3. If 7.5 gal. water weighs 62.4 Ibs., find the weight of 86.5 gal. water.
4. 31 sq. in. is approximately 200 sq. cm. How many square centimeters
in 36.5 sq. in.?
5. If one horse-power is equivalent to 746 watts, how many watts are equiva-
lent to (a) 34.5 horsepower, (b) 5280 horsepower, (c) 0.832 horsepower?
6. If one gallon is equivalent to 3790 cu. em., find the number of gallons of
water in a bottle which contains (a) 4250 cu. em., (b) 9.68 cu. em., (c) 570 cu. em.
or the liquid.
14. The CI and CIF (reciprocal) scales. The reciprocal of a
number is obtained by dividing 1 by the number. ‘Thus, 5 is the
reciprocal of 2, ; (=1+ *) is the reciprocal of 5, and . is the
reciprocal of a.
The reciprocal scales CI, DI and CIF are like the C, D, and CF
scales, respectively, with the exception that they are inverted, i.e.,
the numbers represented by the marks on these scales increase
from right to left. The red numbers associated with the reciprocal
scales enable the operator to recognize these scales. A very im-
portant consideration may be stated as follows: When the hair-
line is set lo a number on the C scale, the reciprocal (or Inverse) of the
number is at the hairline on the CI scale; conversely, when the hair-
line is set to a number on the CI scale, its reciprocal is at the hair-
line on the C scale. A similar relation exists between the D and DI
§14] THE CI AND CIF (RECIPROCAL) SCALES 21
scales and between the CF and CIF scales. If the operator will
close his rule, he can read the opposites indicated in the diagram
ClorDN} 1) 2 | 4 | 5 |
C (or D) | 1 | 0.5 | 0.25 0.125 rn 1111 ma
( = 1/2)| (= 1/4) (= 1/5) (= 1/8)|(=1/9)
By using the facts just mentioned, we can multiply a number
or divide it by the reciprocal of another number. Thus to find
= , we may think of it as 28 x : and
to 28 on D set index of C;
opposite 7 on CI read 4 on D.
Again to find 12 x 3, we may think of it as 12 + : and
3
push hairline to 12 on D,
draw 3 of CI under the hairline;
opposite index of C, read 36 on D.
When the CJ scale is used in multiplicacion and division, the position
of the decimal point is determined in the usual way.
The DF and CIF scales may be used to perform multiplications
and divisions in the same manner as the D and CI scales; thus to
multiply 40.3 by 1/2 04,
to 403 on DF set index of CF;
opposite 904 on CIF, read 4.46 on DF.
Again to multiply 40.3 by 1 /0.207,
to 403 on D set left index of C;
opposite 207 on CIF read 194.7 on DF.
EXERCISES
1. Use the CJ scale to find the reciprocals of 16, 260, 0.72, 0.065, 17.4, 18.5, 67.1.
2. Using the D scale and the CJ scale, multiply 18 by 1/9 and divide 18 by 1/9.
3. Using the D scale and the CJ scale multiply 28.5 by 1/0.385 and divide
28.5 by 1/0.385. Also find 28.5/0.385 and 28.5 x 0.385 by using the C scale and
the D scale.
4. Using the D scale and the CJ scale multiply 41.3 by 1/0.207 and divide
41.3 by 1/0.207.
5. Perform the operations of Exercises 2, 3, and 4 by using the C/JF scale and
the DF scale.
22 THE PROPORTION PRINCIPLE [Cuap. II
15. Proportions involving the CI scale. The CJ scale may be
used in connection with proportions containing reciprocals. Since
1 . 1 1
any number a= 1 + = and since aw ks 1, we have
Rule C. The value of any ratio is not changed if any factor of ils
numerator be replaced by 1 and its reciprocal be written in the denomina-
tor, or if any factor of its denominator be replaced by 1 and its reciprocal
be written in the numerator. Thus ; =a (?) m (i ai Hence
o- +. = = & we
if = ane, we may write sao ap ; if ax = be, we
‘ x b c a a aes
may write (ja) ~ Ul) ~ GP)’ A few examples will indicate
the method of applying these ideas in computations.
To find the value of y which satisfies 7 i = 0.785 x 3.76, apply
D- y__0.785__
Rule C to get Cc: 4.27 (1/3.76)
Since when 3.76 of CJ is under the hairline, 1 /3.76 of C is also under
the hairline
push hairline to 785 on D,
draw 376 of CI under the hairline;
opposite 427 on CF, read y = 12.60 on DF.
The position of the decimal point was determined by observing that
y was near to4 x 1 x 4= 16.
To find the value of y which satisfies 7.89 y= eee use
~ =. Di. y _ 0.0645 ,
Rule (C) to obtain GC? (177.89) ~ 0.381?
and push hairline to 645 on D,
draw 381 of C under the bairline;
opposite 789 on CI, read y = 0.0215 on D.
The position of the decimal point was determined by observing that
.06 is about 4 of 0.38, that y is therefore about £ of 3, or about 0.02.
§16) COMBINED MULTIPLICATION AND DIVISION 238
To find the values of 2 and y which satisfy 57.62 = 0.846y = 7,
use Rule (©) to obtain
a erp oe} (a)
CI’ (1/57.6) 170.846) “1 a
and to 7 on D set index of CI;
opposite 576 on CI, read x = 0.1215 on D;
opposite 846 on CIF, read y = 8.27 on DF.
The folded scales may also be used in this connection. Thus
to solve equations (a),
to 7 on DF set index of CIF;
opposite 576 on CIF, read x = 0.1215 on DF;
opposite 846 on CIF, read y = 8.27 on DF.
EXERCISES
In each of the following equations find the values of the unknown numbers:
=
1. 3.32 = 4.4y = 3a’
2. 76.12 = 3.44y = “a4
3. 1.832 = 5¥, = (162) (1.75).
0.342 y
4, OSE _ = (189) (0.734).
5. 5.832 = 6.44y = ns = 0.2804.
63.42.22 = 8 = (2.78) (13.62).
16. Combined iEain and division.
7.36 X 8.44
92 .
Solution. Reason as follows: first divide 7.36 by 92 and then
multiply the result by 8.44. This would suggest that we
push hairline to 736 on D,
draw 92 of C under the hairline;
opposite 8.44 on C, read 0.675 on D.
18 x 45 x 37
23 xX 29
Solution. Reason as follows: (a) divide 18 by 28, (b) multiply
Example 1. Find the value of
Example 2. Find the value of
24 THE PROPORTION PRINCIPLE [Cuap. II
the result by 45, (c) divide this second result by 29, (d) multiply
this third result by 37. This argument suggests that we
push hairline to 18 on D,
draw 23 of C under the hairline.
push hairline to 45 on C,
draw 29 of C under the hairline,
push hairline to 37 on C,
at the hairline read 449 on D.
To determine the position of the decimal point write
eae = about 50. Hence the answer is 44.9.
A little reflection on the procedure of Example 2 will enable the
operator to evaluate by the shortest method expre