Log Log Duplex Decitrig Slide Rule No. 4081: A Manual (1955 printing)
SLIDE RULE
MANUAL
LOG LOG DUPLEX
DECITRIG
No. 4081
By
LYMAN M. KELLS,
WILLIS F. KERN
and
JAMES R. BLAND
KEUFFEL & ESSER CO.
LOG LOG DUPLEX
DECITRIG®
SLIDE RULE
No. 4081
MANUAL
BY
LYMAN M. KELLs, PH.D.
Professor Emeritus of Mathematics
WILLIS F. KERN
Formerly Associate Professor of Mathematica
AND
JAMES R. BLAND
Profeasor of Mathematics
All at the United States Naval Academy
PUBLISHED BY
KEUFFEL & ESSER CO.
NEW YORK * HOBOKEN, N. J.
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Drafting, Reproduction, Surveying, Optical Tooling
Equipment and Materials, Slide Rules, Measuring Tapes
CopyricHt 1943, 1945, 1947, 1955 By
Keurret & Esser Co.
PRINTED IN VU. 5S. A.
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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, 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 Chapters I and II, they are applied to simple
problems in multiplication and division; in Chapters Lil, IV, and
V they are used to solve problems involving multiplication,
division, square and cube root, trigonometry, logarithms, and
powers of numbers.
Chapter V1 explains the slide rule from the logarithmic stand-
point. 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 I
MULTIPLICATION AND DIVISION
ART. PAGE
La INGHORUCHON: ... « «oy & ws ne ee we ew il eH mw 3
2. Reading the scales ........2..-. SESS eR 3
3. Accuracy of the slide rule ..... ike EKA HS BR Kw e 6
4. Definitions. ..........0.4. Loe 7
6: Multiplication «sme o8 8s wR RE EES 7
6, Either index may be used . ... 2.2... 9
tiDiyision <4 ia,.c%8 S€6 Ri GES a 9
8. Simple applications, percentage, rates . . Skewes a zen 2 FO
9. Use of the scales DF and CF (folded scalens Ce ek 12
CHAPTER II
THE PROPORTION PRINCIPLE AND COMBINED OPERATIONS
10. Introduction. .. ........2.. rn 2)
11. Use of proportions .... SER HRLR ESR eee es TS
12. Forming proportions from equations. (te VHRR ea ee x TD
18. Equivalent expressions of quantity ...... re. |
14. The DJ, CI, and CIF (reciprocal) scales . LR RAM s 22
15. Proportions involving the reciprocal scales. Co : 24
16. Combined multiplication and division . . . . .......... «26
CHAPTER III
SQUARES AND SQUARE ROOTS, CUBES AND CUBE ROOTS
17, SQUATES Sink GREK HARRSK SRE RRS SHR BME se BO
18. Square roots . . « % . 380
19. Evaluation of simple expressions containing square roots ‘and: squares . 82
20. Combined operations involving square roots and squares. . .... 34
22; Cube:rocte:s ¢ i G68 SEE RSE R&B MAS Res 236
28: Combined operations: . c« sme a we Bee we ee ee ey ee ws TOT
24. The Lseale .. 2... ee ee ee ee eee ee 640
CHAPTER IV
TRIGONOMETRY
25. Some important formulas from plane trigonometry ........ 42
26. The S (Sine) and SRT (Sine, Radian, Tangent) scales we & an
27. Simple operations involving the Sand SRT’ scales ......... 46
28. Law of sines applied to solve atriangle ...........2.. «47
29. Short cut in solving atriangle ...........46664.2.24.2. 49
80. The 7 (Tangent) scale. . 2. 2...) 2 we ee ee ee 61
CHAPTER IV (Continued)
ART. PAGE
31. Other functions on the Sand 7 scales... .........2. 52
32. The law of sines applied to right triangles with two legs given ... 53
33. Use of the D/ scale in solving right triangles... 2... 2.2... 55
34. Solution of a triangle for which two sides and the included angle are
CVE c& ag 6 HRBRE TS DRAM ESRB ARE HH wa 56
35. Law of cosines applied to solve triangles for which three sides are
given 2... 59
36. Law of sines applied to oblique triangles, continued. The ambiguous i
COE BER EEE MBER HOKE EE RAK BEB SS FT ees
ST.:Gombined operations: . . «2 6% He we ew Ew we Rw ee 62
38. Radians. Small angles. . . . . ose ime@aktinGae &% P 65
$3. Summary) « ce s a0 we = & WER MARS TR Tew a we w vetlle 68
40. Miscellaneous Exercises . . . . . rn 69
41. Applications involving vectors ... .....-......22504 69
42. Applications ...... ; 73
43. Miscellaneous Exercises . . . 75
CHAPTER V
THE LOG LOG SCALES
AA. Introductionies: Bc GR ECR ETA ROR ER wee HRS «x “TF
45. Mated Seales. . 2. 2... 2. 7
46; Powers Ol@: ss SSE REKRBRSR CARB RMASR SSE 78
47. Powers of any number. Elementary DG ge Ges Bie YaR ee GouNeE we De 82
48. Powers of any number. General ...........2.244 84
49. Logarithms tothe basee. ....... [a Qe wine wwe we h 87
50. Logarithms to any base ............ is & GS HEP BS Me Ke 88
61. Continuous relation of C scales to Log Log scales. . ....... 90
52. The proportion principle for Log Log scales 90
53. Powers-of-ten notation. . . . L 93
54. Numbers less than 0.00005 or greater than 22000 or ee 94
655. Numbers between 0.99 and 1.01. so ee 96
66. Reciprocals. . . 2... 2... . SRS RRBWRES REG SHRS 100
57. Miscellaneous Exercises .. . 2... 10. ee ee tt wt ew 102
CHAPTER VI
LOGARITHMS AND THE SLIDE RULE
58. Construction of the Dscale. . . . 2... 1 1 eee et tte 103
59. Accuracy. .... CC . . . . 104
60. Multiplication and division . LS FS ae EE oe 3 wid
61. The inverted scales... .. . . 105
62. The A scale, the B scale, and the K arale fee ee ee ee 106
63. The trigonometric scales. . . . . fie RO Se ea wT
64. Anapplication . 2... 2... 2) ee ee 110
66. The Log Logacales 2 25 2G HEE PRR MERKR HER SBE S 110
Historical note. . . [EB RR ERC Te RES BH HW 113
Answers. 6. 6 6 2 et 116
INGGX BGK KERR CERES ASR BK ERS Ee swe oe 126
nw
CHAPTER I
MULTIPLICATION AND DIVISION
1. Introduction. This manual is designed to enable any interested
person to learn to use the slide rule efficiently. The beginner should
keep his slide rule before him while reading the manual, 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
7 pe $F | deradl
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 having a rule of different length will be able to understand his rule in
the light of the explanation given.
4 MULTIPLICATION AND DIVISION §2
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 italic'zed number gives the reading to be associated
D 1} I ! | il | Baad Locsie ak: 8 iy Pea |
Fic. 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 1 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
o Pett be £83 sa id i
Scote D
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 ~
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 MULTIPLICATION AND DIVISION 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 Fig. (a). The second
figure from the left, namely 4, tells us that the position lies between
re
7 ;
| 5 ff
Te
lic. a.
the secondary marks associated with 24 and 25. ‘This region is
indicated by the brace in Fig. (6). Now there are 4 marks between
Oo
Cho oP PPP REP RPE pet Prt hy
pt oh kk ob bs bbb ed Pre iriryaey
Fia. 6,
the secondary marks associated with 24 and 25. With these are
associated the numbers 242, 244, 246, and 248 respectively. Thus
246
Fia. c.
the position representing 246 is indicated by the arrow in Fig. (c).
lig. (abe) gives a condensed summary of the process.
A.
246 LIES BETWEEN 240 ANDO 250
vfs 4
Fra. abe.
It 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
gits should be read; the first three will be exact and the last one
TT meme
4 5
Fia. 4.
MULTIPLICATION AND DIVISION
estimated. No attempt should be made to
read more than three digits for positions to
the right of the primary mark numbered 2.
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 following numbers: A 365, B 327, C 263,
D 1745, E 1347, F 305, G 207, H 1078, [ 435,
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.
Assuming that the error of a reading is one
tenth of the smallest interval following the
left-hand index of D, we conclude that the
error is roughly 1 in 1000 or one tenth of one
per cent. The effect of the assumed error
in judging a distance is inversely propor-
tional to the length of the rule. Hence
we associate with a 10-inch slide rule an
error of one tenth of one per cent, with a
20-inch slide rule an error of one twentieth
of one per cent or 1 part in 2000, and with
the Thacher Cylindrical slide rule an error
of a hundredth of one per cent or one part
in 10,000. The accuracy obtainable with
the 10-inch slide rule is sufficient for many
practical purposes; in any case the slide
rule result serves as a check.
§5 MULTIPLICATION AND DIVISION 7
4. Definitions. The central sliding part of the rule is called the
slide, the other part the body. The glass runner is called the indt-
cator 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. The process of multiplication may be per-
formed by using scales C and D. The C seale 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.
wis “ eee ; ‘
is OF yp ppp ere . F '
i TITttTy t ¢ + ; j > ais
Fia. 5.
Fig. 5 shows the rule in skeleton form set for multiplying 2 by 4.
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 for the setting in skeleton form.
€Boecyoy pees
Tt T '
epee
Fia. 6.
8 MULTIPLICATION AND DIVISION §5
To Multiply 1.5 < 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 - ea eee
Se a a toas. Shae ae:
aes Mees.
Fl Z ht aren = ae may
Fia. 7.
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 it is near to3 X 16 = 48. Hence the answer is 47.4.
To find the value of 0.001753 x 12.17,
to 1753 on D set left index of C,
push hairline to 1217 on C,
at the hairline read 213 on D.
To place the decimal point, approximate the answer by writing
.002 X 10 = .02. Hence the answer is 0.0213.
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 of the D scale.
The decimal point is placed in accordance with the result of a rough
calculation.
EXERCISES
1.3 X 2. 6. 1.75 X 5.5. 11. 1.047 X 3080.
2. 3.5 X 2. 7. 4.33 X 11.5. 12. 0.00205 X 408.
3. 5 X 2. 8. 2.03 X 167.3. 13. (3.142)*.
4.2 & 4.55. 9. 1.536 X 30.6. 14. (1.756)?.
5. 4.5 X 1.5 10. 0.0756 & 1.093.
§7 MULTIPLICATION AND 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:
Rule. When a number is to be read on the D scale opposite a num-
ber 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. Then make the desired reading.
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.
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 0.03 600 = 18.
Hence the product is 17.71.
EXERCISES
Perform the indicated multiplications:
1.3 X 5. 9. 912 XK 0.267.
2. 3.05 X 5.17. 10. 48.7 X 1.173.
3. 5.56 & 634. 11, 0.298 0.544.
4. 743 X 0.0567. 12. 0.0456 X 4.40.
5. 0.0495 X 0.0267. 13. 8640 X 0.01973.
6. 1.876 X 926. 14, (75.0).
7. 1.876 X 5.32. 15. (83.0)*.
8. 42.3 X 31.7. 16. 4.98 X 576.
7. Division. The process of division is performed by using the
C and D scales.
To divide 8 by 4 (see Fig. 8),
push hairline to 8 on D,
draw 4 of C under the hairline,
opposite index of C read 2 on D.
10 MULTIPLICATION AND DIVISION §8
a Prpiriiris : ' n
+ t
‘e; Wa) UB i io
Fia. 8.
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. To 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. 9 3.14 + 2.72.
2. 3.75 + 0.0227. 10. 3.42 + 81.7.
3. 0.685 + 8.93. 11, 529 + 565.
4. 1029 + 9.70. 12, 0.0456 + 0.0297.
5. 0.00377 + 5.29. 13. 396 + 0.643.
6. 2875 + 37.1. 14. 0.0592 + 1.983.
7. 871 + 0.468. 15. 0.378 + 0.0762.
8. 0.0385 + 0.001462. 16. 10.05 + 30.3.
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 a number N is N X 1/100; hence 5% of N is
N X 5/100, and, in general, p% of N is 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.
§8 MULTIPLICATION AND DIVISION W
80
Since (83/100) X 1872 is approximately 100 x 2000= 1600, the
answer is 1554.
To find the answer to the question “M is what per cent of NV?” we
must find 100 4 + N. Thus, to find the answer to the question
“87 is what per cent of 184.7?” we must divide 87 & 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.
9000
The rough calculation to * 45 shows that the decimal point should
be placed after the 7. Hence the answer is 47.1%.
For a body moving with a constant velocity, distance = rate times
(ime. Hence if we write d for distance, r for rate, and ¢ for time, we
have
d
d=rfr, orr= or t=-.,
r
’
~iR
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, write r = 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 haver = 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.
12 MULTIPLICATION AND DIVISION §9
to
. 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.
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 D and the C 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 selling
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
| CF l 2 4 5 6 6.67
| ar | ee 3 6
and the same opposites will appear on the C and D scales.
~]
ao
©
_
The following statement relating to the folded scales is basic. The
process of setting the hairline to a number N on scale C to find its
opposite M on scale D may be replaced by setting the hairline lo n
on scale CF to find its opposite m on scale DF. The statement holds
true if letters C and D are interchanged.
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,
89 MULTIPLICATION AND DIVISION 13
to 2 on D set left 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;
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 0.911 on DF;
or
push hairline to 768 on DF,
draw 843 of CF under the hairline,
opposite the index of C read 0.911 on D.
It now appears that we may perform a multiplication or a division
in several ways by using two or more of the scales 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
| limes ils opposite on D, and any number on D is 1/% times its opposite
on DF. For example, to find the value of 47,
push hairline to 4 on D,
at hairline read on DF, 12.57 = 4m,
to find the value ot 2
ae
push hairline to 3 on DF,
at hairline read on D, 0.955 =3/T.
14 MULTIPLICATION AND DIVISION §9
EXERCISES
Perform each of the operations indicated in the following exercises. Whenever
possible without resetting, read the answer on D and also on DF:
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 * 53,600. 12. 783 =.
5. 1.769 + 496. 13. 783 + =.
6. 946 + 0.0677. 14. 0.0876 =.
7. 813 X 1.951. 15. 0.504 + =.
8. 0.00755 + 0.338. 16. 1.072 + 10.97.
17. The circumference of a circle measures 8.43 inches. Find its diameter.
18. A cylindrical tube is 13 inches long and has an outside diameter of 2!
inches. Find its outside surface area.
CHAPTER II
THE PROPORTION PRINCIPLE AND
COMBINED OPERATIONS
10. Introduction. The ratio of two numbers aand b is the quotient
of a divided by bora/b. A statement of equality between two ratios
is called a proportion. Thus
2 6 zx 7 a
3 9° 5 Il’ »b
are proportions. We shall at times refer to equations having such
forms as
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
16 THE PROPORTION PRINCIPLE §
draw 1 of C opposite 2 on D (see Fig. 1) and find the opposites in-
dicated in the following table:
C (or CF)|| 1 1.5 2.5 3 4 5
D(or DF)|| 2 3 5 6 8 10
and draw 2 of C over 1 on D and read the same opposites. The same
statement is true if in it we replace C scale by CF scale and D scale
by DF scale. Hence, if 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.
lor example, let it be required to find the value of x satisfying
zx 9
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. Figure 2 indicates the setting.
oe Ee yes ae ae ~
gi THE PROPORTION PRINCIPLE 7
To find the values of z, y, and z defined by the equations
C 3.15 z 57.6 z
D° 529° 435° y 1834’
note that C and D indicate the respective scales for the numerators
and the denominators, 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.
When an answer cannot be read, apply the italicized rule of §6.
Thus to find the values of x and y satisfying
C x 14.56 5,78
D° 587 97.6 y’
to 976 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 z = 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 x and y satisfying the equations
C(orCF), 492 1 y
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
18 THE PROPORTION PRINCIPLE gu
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:
z 78
l-= .
5 9
9:
120 170
s 4
2 z
‘tii- —_—
3 7.83
‘ 246
“709 oy BSH
7 17 1865 1.6
‘x 8.53
5.28 2.01
10. = = 7 ,
0.204 0.0506 = 30.1034
Ei. sa =
2.85 4.61 u
ix y 2.43
12. ee — = :
0.429 0.789 0.0276
13, —Z_— ~ 0-743 _ 0.0615.
* 0.00560 ! y
14. ey | 3.75
15 ze ef yy _ 1.076
“496 y 3.58 0.287"
§12 THE PROPORTION PRINCIPLE 19
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
la (1)
c
we may write the proportion
x ab
1 c
or we may divide both sides by a to get
x ab x ob
= OF = = =, (3)
aac asec
or we may multiply both sides by c/x to obtain
cx cab cab
—=—, r-=-—. (4)
x xe 1 x
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 writlen 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,
16 28 16 X 28
For example, to find — bse: , write z = ———— , apply Rule
35 35
C x 28
B btain —: ==>
( ) to obtain = 16 ~ 38’
and push hairline to 35 on D,
draw 28 of C under the hairline,
opposite 16 on D, read x = 12.8 on C.
Figure 3 indicates the setting.
OL ste peeseret ti lal + tg
eT TE See spe abe lt.
Fic. 3.
20 THE PROPORTION PRINCIPLE §12
To recall the rule for dividing a given number M by a second given
M
M
number N, write x = —, apply Rule (A) to obtain >: ; a
N
and push hairline to M on D,
draw N of C under the hairline,
opposite index of C, read x on D.
V4
< , apply Rule(B)
To recall the rule for multiplication, set z =
_ Dd x N
to obtain = : M71?
and to N on D set index of C,
opposite M on C, read x on D,
: see 7.48 864
wel is B — =
To find 2 if © = Gar ogg)’ wee Rule (B) to get“ = oe,
make the corresponding setting and read x = 0.221. The position of
the decimal point was determined by observing that + must be about
as of 8, or 0.2.
EXERCISES
Find in each case the value of the unknown quantity:
8 xX 12 89.32
om 8. 498 = ;
om 7 3 0563
3.95 X 0.707
5 Tiiw ee, £0,
28 x
3. 8y = 75.6 X 9. 10, 0.608 « 2:2879
4 y = 80K 108 4
ye 18 Le 0.772
‘ 147.5 X 8.76 3860 2.8by
oe 3260. 12. 2580y = 17.9 X 587.
0.797 X 5.96 13. 3.14y = 0.785 X 38.7.
y= —————.
0.502 0.876y
7. = 75.7. ,
§13 THE PROPORTION PRINCIPLE 21
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. =z
144 no. of sq. in. 3210’
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, zt
6080 no. of ord. mi. 28.5’
make the corresponding setting and read x = 24.8 naut. mi.
EXERCISES
1. An inch is equivalent to 2.54 cm. Find the respective length in cm. of rods
66 in. long, 98 in. long, and 386 in. long. Note the proportion:
in. 1 66 98 386
om. 254 2 y 2.
2. One yd. is equivalent to 0.9144 meters. Find the number of meters in a
distance of (a) 300 yd. (b) 875 yd. (c) 2.78 yd.
yd
,
1_ _ 300 _ 875 _ 2.78
Hint.
m 0.914. y z
3. If 7.5 gal. water weighs 62.4 lbs., find the weight of (a) 86.5 gal. water.
(b) 247 gal. water. (c) 3.78 gal. water.
4. 31 sq. in. is approximately 200 sq. cm. How many square centimeters in
(2) 36.5 sq. in.? (b) 144 sq. in.? (c) 65.3 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. cm. (b) 9.68 cu. em. (c) 570 cu. em.
of the liquid.
*A table of equivalents is included with each K & E slide rule.
22 THE PROPORTION PRINCIPLE §14
7. The intensity of pressure due to a column of mercury 1 inch high (1 inch
of mercury) is 0.49 Ib. per sq. in. If atmospheric pressure is 14.2 lb. per sq. in.,
what is atmospheric pressure in inches of mercury? What is a pressure of 286
Ib. per sq. in. in inches of mercury? What is a pressure of 128 inches of mercury
in Ib. per sq. in.?
8. If P; represents the pressure per square unit on a given quantity of a
perfect gas and V, the corresponding volume, then for two states of the gas
at the same temperature
P, V2,
P, V;
The volume of a gas at constant temperature and pressure 14.7 lb. per sq.
in. is 125 cu.in. (a) Find the respective pressures at which the volumes of
the gas are 300 cu. in., 250 cu. in., 75.0 cu. in. (hb) Find the respective vol-
umes of the gas under the pressures: 85 lb. per sq. in., 55 1b. per sq. in., 23 Ib.
per sq. in, 10 Ib. per sq. in.
14. The DI, Cl, and CIF (reciprocal) scales. The reciprocal of a
1
number is obtained by dividing 1 by the number. Thus, 7 is the
3 3 1
(=1+ 5) is the reciprocal of 3° and ~ se the
“ a
wi bo
reciprocal of 2,
reciprocal of a.
The reciprocal scales DI, CI, and CIF are marked and numbered
like the D, C, and CF scales respectively but in the reverse (or in-
verted) order; that is, 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 important consideration may be stated as follows: When the
hairline is set to 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 ts set to a number on the CI scale, its reciprocal is at the hairline
on the C scale.
The same relation exists between the D and DI scales and between
the CF and CIF scales.
To fix this relation in mind push the hairline in succession to the
| | 0.5 | 0.25 0.2 0.125 | 0.1111
| | (=1/2) (=1/4) | (=1/5) | (=1/8) | (=1/9) |
| pr | 1! 2 !| 4 5 8 9 |
numbers on D/ in the second row of the diagram and read on D the
respective opposites written in the first row. Also opposite the
§14 THE PROPORTION PRINCIPLE 23
numbers 1, 2, 4, 5, 8, and 9 on CJ read their respective reciprocals
on C. Again find the same opposites on CIF and CF.
By using the facts just mentioned, we can multiply a number
or divide it by the reciprocal of another number. Thus to find
28 . . 1
=, we may think of it as 28 X = and
‘ ‘
to 28 on D set index of C,
opposite 7 on CJ read 4 on D.
Again to find 12 & 3, we may think of it as 12 + : and
push hairline to 12 on D,
draw 3 of CJ under the hairline,
opposite index of C, read 36 on D.
When the CJ scale is used in multiplication and division, the positior
of the decimal point is determined in the usual way.
The DF and CIF scales may be used to perform multiplication:
and divisions in the same manner as the D and C7 seales; thus t
multiply 40.3 by 1/9.04,
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 DJ scale to find the reciprocals of 16, 260, 0.72, 0.065, 17.4, 18.5, 67.1
2. Find 18.2 X 21.7 in the usual way and then read 1/(18.2 x 21.7) on D)
opposite the first answer on D. Similarly find the values of 1/(2.87 x 623),
and 1/(0.324 0.497).
3. Using the D scale and the CJ scale, multiply 18 by 1/9 and divide 18 by 1/9.
4. Using the D scale and the CJ seale 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.
5. Using the D seale and the CJ scale multiply 41.3 by 1/0.207 and divide
41.3 by 1/0.207.
6. Perform the operations of Exercises 2, 3, and 4 by using the CJF scale and
the DF scale.
7. Set the hairline to 8.62 on DF, and read at the hairline 8.62/< on D and
*/8.62 on DJ. Also find the values of 1.23/z, =/1.23, 39.4/x, and =/39.4.
24 THE PROPORTION PRINCIPLE §15
15. Proportions involving the reciprocal scales. The reciprocal scales
may be used in connection with proportions containing reciprocals.
1 1 1
Since any number a = 1 + — and since nt + 1, we have
a . a
Rule (C). The value of any ratio is not changed if any factor of its
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
1
be writlen in the numerator. Thus 5 = 4 ) = b a Hence
b
if = bc, we may write = = (jo an nD if ax = bc, we
co b _ ¢
(1/a) (1/e) —(1/b) *
the method of applying these ideas in computations.
may write A few examples will indicate
To find the value of y which satisfies / _ 0.785 3.76, apply
4.27
D. y _ _0.785
Rule (C) to get C’ 427 (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 1/(7.89y) = 0.381/0.0645, use
_D. (1/y) _ 0.381
Rule (C) to obtain: O/0 — S551. and
push hairline to 381 on D,
draw 645 of C under the hairline,
opposite 7.89 on C read y = 0.0215 on DI.
The position of the decimal point was obtained by observing that
0.381 is about 6 X 0.06 and therefore that 1/y is about 6 X 7.89, or
y = 0.02 approximately.
To find the values of x and y which satisfy 57.62 = 0.846y = 7,
use Rule (C) to obtain
D. x y 7
CI’ (1/57.6) (1/0.846) 1’ (2)
§16 THE PROPORTION PRINCIPLE 25
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 equation (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:
75.2 0.342 y
1. 3.382 = 4.4y = ——- 4. = ——-s 0.734).
* 4 7.342 . ae
111 :
2. 76.12 = 3.444 = —- 5. 5.832 = 6.44y = BS = 0.2804.
22.8 £
1.83 17.6
3. 1.882 = —— = (162) (1.75). 6. 3.422 =—— = —— = (2,78) (13.62).
24.5 y z
16. Combined multiplication and division.
Example 1. Find the value of ee
Solution. Reason as follows: first divide