RF Cascade Workbook

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Slide
rules were an enigma and a thing to be feared to most people even in the days when handheld digital calculators
had not entered the scene. The rows of numbers generally look nothing like a simple, familiar ruler, and the sliding
window thingy with the thin line sent kids and adults alike running for the tall grass for cover. As with most things
not too complicated, learning to use the slide rule can be mastered with a little instruction. You don't necessarily
need to understand logarithms and trigonometry functions, but it certainly helps if you also want to understand
how the device works. It is the same thing as not needing to know how your Casio digital calculator works in order
for it to be useful. If you do an Internet search for instructions on slide rule usage, there is no shortage of
documents. This one from Lawrence I believe does a particularly good job because it is filled with examples of the
most common types of mathematical operations, including powers and roots. It does not address trig functions because,
like many beginner's slide rules, it only hade one functional side, with instructions on the reverse side.

See a large selection of slide rules at the Slide Rule Museum from Lawrence, including specialty types for electrical engineering, music, photography, carpentry, machining, model railroading, printing, water pipe fitting, and much more. Here is the lowly RF Cafe Slide Rule & Calculator Museum.

**The Quick and Easy "Lawrence"
Slide Rule Instruction Book**

**SIMPLE RULES for using the SLIDE RULE **

By John Poland

Prof. of Mechanical Engineering and Air Conditioning at the Chicago Technical College

**FOREWORD **

Anyone from a seventh grade student up can learn to use a slide rule properly with just a little study and practice.

There are two methods of learning to use the slide rule:

1. The Mental Survey Method

2. The Digit and Integral Digit Method

The Mental Survey method is quick and practical. It is used by hundred of, thousands of slide rule operators. The Digit and Integral Digit method is more thorough and scientific. It is used In schools and colleges.

To the best of our knowledge this is the first time both methods have been made available in one instruction book.

If you select .the "Mental Survey" 'method disregard all material on digits and integral digits. If you use the Digit and Integral Digit method, disregard all material on the "Mental Survey method.

Anyone who will apply himself to the operation of the slide rule and will make its use a practical habit while at work, study or leisure, will find himself richly rewarded in the saving of time and energy - to say nothing of the satisfaction that comes rom the mastery of "Man's most useful tool."

Published for

ENGINEERING INSTRUMENTS, INC.

PERU, INDIANA

1952

COPYRIGHT 1939 BY LAWRENCE ENGINEERING SERVICE - PERU, IND.

COPYRIGHT 1940 BY LAWRENCE ENGINEERING SERVICE - PERU, IND.

INTERNATIONAL COPYRIGHT SECURED

**THE SLIDE RULE **

The Slide Rule is a time and labor saving instrument for quickly solving problems involving multiplication, division, proportion, squares, square roots, cube and cube roots or any combination of these processes.

The operation of a Slide Rule is simple and anyone with a fair understanding of numbers can use one. It should be remembered that the Rule itself is accurate within about one-half of one percent, and the accuracy of the answers obtained are limited only by the spacing of the lines on the scales and the ability of the user to estimate readings and setting that fall between the lines.

It must also be remembered that only the first three digits of any number can be read or located on the slide rule except in Units 1 of the C and D scales, where four digits can be located. A good rule to follow is this: if the fourth digit is less than 5, drop the remaining digits but if the fourth digit is 5 or more, increase the third digit by one and drop the remaining ones.

The Slide Rule consists of three parts (See Fig. 1). namely: the rule proper or body containing the A, D and K scales; the slide containing scales B, C I and C; and the glass "indicator," "runner," or. "cursor" with a hair line in the center.

**Scales and Uses**

Scales C and D are used in multiplication, division and their combinations. Scales A and D are used for squaring and for finding the square roots of numbers. Scales K and D are used for cubing and finding the cube root of numbers. The C I scale gives the reciprocal of any number on the scale C. It can be observed from Fig 1 that the scales A and B are identical, scales C and D are also identical and that the C I scale is the C scale in reverse direction.

Before the different operations of multiplication, division, etc., can be accomplished, the operator must know how to read the different scales.

**Scales "C" and "D"**

From Fig. 1 it will be seen that these two scales are identical in their divisions and markings. Since most of the operations are based upon scale D we will learn to read this one first, remembering that the discussion to follow can be applied to scale C. Looking at scale D in Fig. 1 we see that it is divided into nine main unequal divisions, starting at the left with number 1 and proceeding to the right to number 9, thence to a second number 1.

These divisions as mentioned before are UNIT divisions and each one is divided into smaller divisions. From Fig. 1 we see that the UNIT 1 (that is from number 1 to number 2 has ten major divisions, each equal to one-tenth (.1) of the unit. These major divisions in turn are divided into ten divisions, each of which is one-tenth (.1) of the major division of one-hundredth (.01) of the UNIT

Examples: 17. 15.3. 1162. and 1.92 would be located as shown in Fig. 2.

UNIT ONE OF SCALE "D"

Each of these numbers begin with the digit "one" so each will fall in UNIT 1 of scale D. Number 17 is read at unit 1 plus 7 major divisions: number 15.3 is read at unit 1 plus 5 major divisions plus 3 minor divisions of the following major division; number 1.92 is read at unit 1 plus 9 major divisions plus 2 minor divisions of the following major division. It should be noted that the decimal point is always disregarded when locating a number on the scale.

UNIT 2: Unit 2 is the next main division of scale D and from Fig. 1 we see it is divided into ten major divisions as is unit 1 but they are not numbered here. Each of the major divisions of this unit are divided into 5 minor divisions, each of which is one-fifth of two-tenths (.2) of the major division or (.02) of UNIT 2

Examples of UNIT 2: Locate numbers 21.6, .283 and 255. (S Fig. 3.)

Each of these numbers begins with the digit 2 so it will fall within unit 2. The number 21.6 is read at unit 2 plus 1 major division plus 3 minor divisions of the following major division as each line here equals two-tenths (.2); the number .283 is read at unit 2 plus 8 major divisions plus 1 1/2 minor divisions of the following major division; the number 255 is read at unit 2 plus 5 major divisions plus 2 1/2 minor divisions of the following major division. Here again the decimal point is disregarded when locating a number on the scale.

UNIT 3: Unit 3 of the D scale is divided as is unit 2 and the divisions possess the same values. See Fig. 3 for comparing unit 3 and unit 2.

Examples for UNIT 3. Locate numbers 3.08. 37.1 and 359. (See Fig. 3.)

The number of 3.08 is read unit 3 plus 0 major divisions plus 4 minor divisions of the first major division; the. number 3.71 is read unit 3 plus 7 major divisions plus 1/2 of a minor division of the following major division; the number 359 is read unit 3 plus 5 major divisions plus 4 1/2 minor divisions of the following major division.

UNIT 4: Unit 4: is our next main division and from Fig. 1 we see that this unit is divided into ten major divisions as in the preceding units. Each of these major divisions is in turn divided into two minor division divisions each having a value of one-half or .5 of the major division or .05 of unit 4. When we have a three digit number in this unit the third digit must be estimated unless it is a 5.

Examples: Number 435 is read unit 4 plus 3 major divisions plus 1 minor division; number 43716 is read unit 4 plus 3 major divisions plus 1 minor division which gives 435 and then add an estimated two-fifths of the next minor division giving 437. The remaining digits, 1 and 6, are too small to be considered so they are ignored as explained under the portion headed "THE SLIDE RULE." See Fig. 4 for these and other readings.

The remaining units on the scale are the same as unit 4 except that the division lines are closer together. Examples for units 5, 6, 7, 8 and 9 are shown in Fig. 5.

The number of 5.32 is read unit 5 plus 3 major divisions plus an estimated two-fifths of the next minor division; number .808 is read unit 8 plus 0 major divisions plus 1 minor division which gives 805 and then add three-fifths of the next minor division giving 808; number 9622 is read unit 9 plus 6 major divisions plus two-fifths of the next minor division. (The fourth digit 2 is omitted on the scale as previously explained.)

**LOCATION OF DECIMAL POINT IN MULTIPLCATION BY THE MENTAL SURVEY METHOD **

**Multiplication**

As stated before, scales C and D are used in multiplication and are identical. Therefore. the divisions and reading of scale C will be the same as those found on scale D. The first line. number 1, at the left of the C scale is called the left index, while the last line at the right end of the, scale is called the right index.

When multiplying two numbers together we set either the right or the left index of the C scale directly over one of the numbers on the D scale; move the indicator so the hair line is over the number on the C scale; then read the answer under the hair line on the D scale.

Example 1: 3.4 x 7.2

Set the right index of the C scale over 7.2 on the D scale, move the indicator until the hair line is over 3.4 on the C scale. Read the 24.5 under the hair line on the D scale.

The most convenient way to locate the decimal point is to make a mental multiplication of the first digits in the factors. Then place the decimal point in the result so that the value is nearest that of the mental multiplication. Thus, 3 and 7, the first digits of our factors will give 21, indicating that the result 245 will become 24.5 instead of 2.45 or 245, as 24 is the nearest to our mental multiplication value of 21.

Example 2: 14 x 16 = 644

Set the left index of the C scale over 14 on the D scale, move the indicator so the hair line is over 46 on the C scale. Read the answer 644 under the hair line on the D scale. (See Fig. 6.)

From mental multiplication we see that 14, which is near 10, and 46, which is near 45, would give 10 x 45 or 450, showing that the answer would be in the hundreds, namely 644 instead of 64.4 or 6440.

Example 3: 18.72 x .356 = 6.66

Set the left index of the C scale over 1872 on the D scale, move the indicator until the hair line is over 356 on the C scale. Read the answer 666 under the hair line on the D scale. (See Fig 8.).

From observation we see that we are multiplying practically 20 x .3 giving 6. Therefore, our answer becomes 6.66 instead of 66.6 or .666.

Multiplication by the Integral Digit Method. (See page 21.)

As stated before scales C and D are used in multiplication and are identical. Therefore. the divisions and readings of scale C will be the same as those found on scale D. The first line, at number 1 at the left of the C scale, is called the left index, while the last line at the right end of the scale is called the right index.

When multiplying two numbers together we set either the right or the left index of the C scale directly over one of the numbers on the D scale; move the indicator so the hair line is over the other number on the C scale; then read the answer under the hair line on the D scale.

Example 1: 2 x 3 = 6

Set the slide so that the left index or number 1 of the C scale is directly over UNIT 2 on the D scale. Move the indicator so that the hair line is over UNIT 3 on the C scale, then read the answer 6 directly under the hair line on the D scale. (See Fig. 7.)

Example 2: 14 x 46 = 644

Set the slide so the left index of the C scale is over 14 on the D scale. Move the indicator so the hair line is over 46 on the C scale, then read the answer 644 under the hair line on the D scale. (See Fig.6.)

It will be noted that the slide extended to the right in both of these operation and that the number of integral digits in each answer contained one less integral digit than the sum of the integral digits in both numbers. From this explanation we can establish a rule for multiplication when the slide extends to the right.

RULE: When the slide extends to the right, add the number of integral digits in the multiplier to those in the multiplicand and then subtract one; the result will be the number of integral digits in the answer.

Example 3: 18.72 x 0.356 = 6.66

Set the slide so that the left index of the C scale is over 187 on the D scale. Move the indicator so that the hair line is over 356 on the C scale, then read the answer 666 under the hair line on the U scale. (See Fig. 8.)

The number 18.72 has plus 2 integral digits and 0.356 has zero integral digits; adding we get plus 2·integral digits. Since the slide extended to the right we subtract 1 leaving plus one integral digit, thus 666 becomes 6.66.

Example 4: 2351 x 41.2 = 96800

The left index of the C scale is set over 235 on the D scale; the hair line of the indicator over 412 on the C scale; read the number 968 under the hair line on the D scale. Number 2351 has 4 integral digits and number 41.2 has 2 integral digits; adding we obtain 6 integral digits. Since the slide moved to the right we subtract one leaving 5 integral digits in the answer. As we are only able to locate three numbers on the scale (three numbers are all that we can read in any answer), zeros must be added to complete the necessary number of integral digits. We will need two in this example making tire answer 96.800.

Problems:

183.2 x 27.42 Ans. 5020

31.52 x 19.7 Ans. 621

1.248 x 6.57 Ans. 8.20

257 x 0.313 Ans. 80.4

0.037 x 228 Ans. 8.44

233.42 x 0.00196 Ans. .457

It will be noted that the preceding problems in multiplication were such that the slide always extended to the right. We will now use numbers so that the slide will move to the left.

Example 1: 7 x 6 = 42

Set the right index of the C scale over 7 on the D scale, move the indicator so that the hair line is over 6 on the C scale, read the answer 42 under the hair line on the D scale. (See Fig 9.)

Example 2: 8.3 x 27 = 224

Set the right index of the C scale over 83 on the D scale, move the indicator so that the hair line is over 27 on the C scale, read the answer 224 under the hair line on the D scale. (See Fig 10.)

Example 3: 9.5 x 4.8 = 45.6

In the above example the slide extended to the left and the answer has as many integral digits as there were in the numbers themselves. Hence the rule: WHEN THE SLIDE EXTENDS TO THE LEFT, ADD THE NUMBER OF INTEGRAL DIGITS IN THE MULTIPLIER TO THOSE IN THE MULTIPLICAND AND THIS SUM WILL BE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER.

Example 4: 75.3 x 6.27 = 472

Number 75.3 has 2 integral digits and 6.27 has one integral digit, adding, we will have 3 integral digits in the answer: thus 472.

Example 5: 4387 x 0.0372 = 163

Number 4387 has 4 integral digits and 0.0372 has minus 1 integral digits, adding 4 to -1 we have 3 integral digits, thus the answer 163.

Example 6: 0.0057 x 0.0244 = .000139

Number 0.0057 has minus 2 (-2) integral digits and 0.0244 has minus 1 (-1) integral digit. Adding -2 to -1 we have -3 integral digits. Therefore, three zeros must be placed before number 138 making the answer 0.000138.

Problems:

75.21 x 485 Ans. 36.500

0.0672 x 3.48 Ans. 0.234

8.052 x 44.72 Ans. 0.360

9.122 x 0.1488 Ans. 1.357

When multiplying more than two numbers as 6 x 24 x 32, multiply 6 x 24 as has been explained; then take this answer and multiply it by 32 to obtain the final answer. Multiplying 6 x 24 = 144. 144 x 32 = 4610.

In the first operation, the slide extended to the left, therefore the answer will have 3 integral digits since there is 1 integral digit in number 6 and there are 2 integral digits in number 24. In the next operation, the slide extends to the right, so we must use the sum of the integral digits less one. Number 144 (answer of first operation) has 3 integral digits and number 32 has 2 integral digits. Adding we have 5 integral digits and less one leaves 4 integral digits in our final answer. As we were only able to read three numbers on the scale we must add 1 zero to make the necessary 4 integral digits.

Problems:

14 x 35 x 8.4 Ans. 4120.

3.76 x 0.786 x 185 Ans. 547.

27.5 x 12.5 x 0.27 Ans. 92.8

**DIVISION **

Location of Decimal Point in Division by the Mental Survey Method

In division we use the same scales, namely C and D, as were used in multiplication, only in reverse order.

When dividing, the hair line of the indicator is placed over the number to be "divided into" (numerator) on the D scale; the slide is moved to the right or left so that the number we are "dividing by" (denominator) is located on the C scale will fall directly under the hair line. The answer will be found under the index of the C scale on the D scale.

Example 1: 8 ÷ 4 = 2

Set the hair line of the indicator over 8 on the D scale, move the slide until 4 on the C scale is under the hair line; read the answer 2 under the left index on the D scale. (See Fig. 11.)

Example 2: 845 ÷ 25 = 33.8

From a mental survey we can see we have practically 800 ÷ 20, giving us two places in the answer; thus we obtain 33.8 instead of 338 or 3.38; or we can determine the decimal point in the following manner; 25 will go into 84 one digit and some left over; 25 will go into the left over amount and 5 another digit, thus giving 2 digits in our answer, so we have 33.8. The setting of the scale is shown in fig. 13.

Example 3:, 6187 ÷ 24.8 = 249

Here 24 will go into 61 one digit and give a remainder; this remainder and 8 can be divided by 24 making another digit in the answer and a remainder. Again this remainder and 7 can be divided by 24 making a third digit in the answer; thus we have three digits, 249 as the answer. Or we can say we have roughly 6000 ÷ 24, giving three numbers or digits in the answer.

**Division
by the Integral Digit Method.**

In division we use the same scales as are used in multiplication, namely: the C and D scales because division is the reverse operation to multiplication.

When dividing two numbers the hair line of the indicator is placed over the numerator or dividend on the D scale; the slide is moved to the right or left so that the denominator or divisor located on tile C scale will fall directly under the hair line; read the answer under the index of the C scale on the D scale.

Example 1: 8 ÷ 4 = 2

Set the hair line of the indicator over 8 on the D scale; move the slide until 4 on the C scale is under the hair line; read the answer 2 on the D scale under the left index of the C scale. (See Fig. 11.)

Example 2: 48 ÷ 3 = 16

Set the hair line of the indicator over 48 on the D scale, move the slide until 3 of the C scale is under the hair line, read the answer 16 on D scale under the left index of the C scale. (See Fig. 12.)

Example 3: 845 ÷ 25 = 33.8

Set the hair line of the indicator over 845 on the D scale, move the slide until 25 of the C scale is under the hair line, read the answer 33.8 on the D scale under the left index of the C scale. (See Fig. 13.)

The user will note that in the preceding examples the slide moved to the left. When this occurs the rule for finding the number of integral digits in the answer is as follows:

IN DIVISION, WHEN THE SLIDE MOVES TO THE RIGHT, SUBTRACT THE NUMBER OF INTEGRAL DIGITS IN THE DIVISOR FROM THE NUMBER OF INTEGRAL DIGITS IN THE DIVIDEND AND THEN ADD 1. THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER.

Referring to example No.1, the dividend 8 has one integral digit and the divisor has one integral digit; subtracting the one integral digit in the dividend leaves zero, but since our slide moves to the right, we add one making 0 + 1 = 1. Therefore. our answer will have one integral digit.

Using example No.2, the dividend 48 has two integral digits and the divisor 3 has one integral digit. Subtracting one from two and then adding 1 we obtain two integral digits in the answer.

With example No.3, the dividend 845 has three integral digits and the divisor 25 has two integral digits. Subtracting two from three and then adding 1 we obtain two integral digits in the answer.

Problems:

72.6 ÷ 1.83 Ans. 39.7

6187 ÷ 24.8 Ans. 249

5.493 ÷ 37 Ans. 0.1484

813.6 ÷ 672 Ans. 1.21

Example 4: 763 ÷ .027 = 28,200

Set the hair line over 763 on the C scale, move the slide until 27 is under the hair line, read the number 282 on the D scale under the index of the C scale. The number 763 has three integral digits, and .027 has minus one integral digit. Since we are dividing, we subtract the integral digit of the divisor from those in the dividend as stated in the rule. But here we are subtracting a minus one from a plus three, which will give four as explained under "Subtraction of Integral Digits." Also the slide extended to the right, so we must add 1 to the result just obtained by subtracting the integral digits, making 4 + 1 or 5 integral digits in our answer. Therefore. our answer 28200; adding the two zeros to make the necessary five integral digits. (See Fig.14.)

Example 5: .0851 ÷ .0362 = 2.35

Set the hair line over 851 on the D scale, move the slide until 362 is under the hair line, read 235 on the D scale under the index. Here the dividend has a minus one integral digit and the divisor also has a minus one integral digit. Subtracting minus one from minus one we get zero, but the slide goes to the right so we must add one to zero making one integral digit in the answer. Therefore. the reading 235 becomes 2.35. (See Fig. 15.)

Example 6: 3852 ÷ 725 = 5.31

Example 7: 143.7 ÷ 9.36 = 15.35

In the examples above, 3852 has four integral digits while 725 described except that the slide will move to the left and the answer will be found under the right index of the C scale. When the slide moves to the left we have the following rule: IN DIVISION, WHEN THE SLIDE MOVES TO THE LEFT, SUBTRACT THE NUMBER OF INTEGRAL DIGITS IN THE DIVISOR FROM THE NUMBER OF INTEGRAL DIGITS IN THE DIVIDEND AND THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER.

In the example above, 3852 has four integral digits while 725 has three. Subtracting, we have one integral digit in the answer, giving 5.32. In example 7, 143.7 has three integral digits while 9.36 has one. Subtracting one from three gives two integral digits in the answer, so we have 15.36.

Example 8: 26.8 -÷ 0.0652 = 411

Here 26.8 the dividend, has two integral digits and .0652, the divisor, has minus one integral digit. Subtracting minus one from two we obtain three, which is the number of integral digits in the answer.

In this problem the rule of subtraction of integral digits is used, as previously explained.

Problems:

4873 ÷ 562 Ans. 8.67

93.14 ÷ 472 Ans. 0.197

0.0568 ÷ 0.027 Ans. 2.1

6.35 ÷ 18.2 Ans. 0.349

Scale A.

Scale A is divided into two halves, called the left and the right half. (See Fig. 1.)

The left half is divided into 9 unequal divisions, similar to scale D, and these divisions are also called UNIT divisions.

Unit 1 is divided just as unit 2 is on the D scale, so the divisions will have the same value as did unit 2 of the D scale. Units 2, 3 and 4 are like unit 4 of the D scale and they have the same value as did unit 4 of the D scale. Unit 5 of the A scale is divided in ten major divisions only, so the third digit of any number must be estimated. This is also true of units 6, 7, 8 and 9.

The right half of the A scale is divided as in the left half, therefore the values of the divisions will be the same.

As stated before, scales A and B are identical, as the readings on scale B will be the same as the reading on scale A.

Continue with Part 2

Posted 5/20/2013

See a large selection of slide rules at the Slide Rule Museum from Lawrence, including specialty types for electrical engineering, music, photography, carpentry, machining, model railroading, printing, water pipe fitting, and much more. Here is the lowly RF Cafe Slide Rule & Calculator Museum.

Kraft Systems advertisement with Slide Rule in February 1972 edition of American Aircraft Modeler |

By John Poland

Prof. of Mechanical Engineering and Air Conditioning at the Chicago Technical College

Anyone from a seventh grade student up can learn to use a slide rule properly with just a little study and practice.

There are two methods of learning to use the slide rule:

1. The Mental Survey Method

2. The Digit and Integral Digit Method

The Mental Survey method is quick and practical. It is used by hundred of, thousands of slide rule operators. The Digit and Integral Digit method is more thorough and scientific. It is used In schools and colleges.

To the best of our knowledge this is the first time both methods have been made available in one instruction book.

If you select .the "Mental Survey" 'method disregard all material on digits and integral digits. If you use the Digit and Integral Digit method, disregard all material on the "Mental Survey method.

Anyone who will apply himself to the operation of the slide rule and will make its use a practical habit while at work, study or leisure, will find himself richly rewarded in the saving of time and energy - to say nothing of the satisfaction that comes rom the mastery of "Man's most useful tool."

Published for

ENGINEERING INSTRUMENTS, INC.

PERU, INDIANA

1952

COPYRIGHT 1939 BY LAWRENCE ENGINEERING SERVICE - PERU, IND.

COPYRIGHT 1940 BY LAWRENCE ENGINEERING SERVICE - PERU, IND.

INTERNATIONAL COPYRIGHT SECURED

The Slide Rule is a time and labor saving instrument for quickly solving problems involving multiplication, division, proportion, squares, square roots, cube and cube roots or any combination of these processes.

The operation of a Slide Rule is simple and anyone with a fair understanding of numbers can use one. It should be remembered that the Rule itself is accurate within about one-half of one percent, and the accuracy of the answers obtained are limited only by the spacing of the lines on the scales and the ability of the user to estimate readings and setting that fall between the lines.

It must also be remembered that only the first three digits of any number can be read or located on the slide rule except in Units 1 of the C and D scales, where four digits can be located. A good rule to follow is this: if the fourth digit is less than 5, drop the remaining digits but if the fourth digit is 5 or more, increase the third digit by one and drop the remaining ones.

The Slide Rule consists of three parts (See Fig. 1). namely: the rule proper or body containing the A, D and K scales; the slide containing scales B, C I and C; and the glass "indicator," "runner," or. "cursor" with a hair line in the center.

**FIG. 1**

Scales C and D are used in multiplication, division and their combinations. Scales A and D are used for squaring and for finding the square roots of numbers. Scales K and D are used for cubing and finding the cube root of numbers. The C I scale gives the reciprocal of any number on the scale C. It can be observed from Fig 1 that the scales A and B are identical, scales C and D are also identical and that the C I scale is the C scale in reverse direction.

Before the different operations of multiplication, division, etc., can be accomplished, the operator must know how to read the different scales.

From Fig. 1 it will be seen that these two scales are identical in their divisions and markings. Since most of the operations are based upon scale D we will learn to read this one first, remembering that the discussion to follow can be applied to scale C. Looking at scale D in Fig. 1 we see that it is divided into nine main unequal divisions, starting at the left with number 1 and proceeding to the right to number 9, thence to a second number 1.

These divisions as mentioned before are UNIT divisions and each one is divided into smaller divisions. From Fig. 1 we see that the UNIT 1 (that is from number 1 to number 2 has ten major divisions, each equal to one-tenth (.1) of the unit. These major divisions in turn are divided into ten divisions, each of which is one-tenth (.1) of the major division of one-hundredth (.01) of the UNIT

Examples: 17. 15.3. 1162. and 1.92 would be located as shown in Fig. 2.

UNIT ONE OF SCALE "D"

**FIG. 2 **

Each of these numbers begin with the digit "one" so each will fall in UNIT 1 of scale D. Number 17 is read at unit 1 plus 7 major divisions: number 15.3 is read at unit 1 plus 5 major divisions plus 3 minor divisions of the following major division; number 1.92 is read at unit 1 plus 9 major divisions plus 2 minor divisions of the following major division. It should be noted that the decimal point is always disregarded when locating a number on the scale.

UNIT 2: Unit 2 is the next main division of scale D and from Fig. 1 we see it is divided into ten major divisions as is unit 1 but they are not numbered here. Each of the major divisions of this unit are divided into 5 minor divisions, each of which is one-fifth of two-tenths (.2) of the major division or (.02) of UNIT 2

Examples of UNIT 2: Locate numbers 21.6, .283 and 255. (S Fig. 3.)

**FIG. 3**

Each of these numbers begins with the digit 2 so it will fall within unit 2. The number 21.6 is read at unit 2 plus 1 major division plus 3 minor divisions of the following major division as each line here equals two-tenths (.2); the number .283 is read at unit 2 plus 8 major divisions plus 1 1/2 minor divisions of the following major division; the number 255 is read at unit 2 plus 5 major divisions plus 2 1/2 minor divisions of the following major division. Here again the decimal point is disregarded when locating a number on the scale.

UNIT 3: Unit 3 of the D scale is divided as is unit 2 and the divisions possess the same values. See Fig. 3 for comparing unit 3 and unit 2.

Examples for UNIT 3. Locate numbers 3.08. 37.1 and 359. (See Fig. 3.)

The number of 3.08 is read unit 3 plus 0 major divisions plus 4 minor divisions of the first major division; the. number 3.71 is read unit 3 plus 7 major divisions plus 1/2 of a minor division of the following major division; the number 359 is read unit 3 plus 5 major divisions plus 4 1/2 minor divisions of the following major division.

UNIT 4: Unit 4: is our next main division and from Fig. 1 we see that this unit is divided into ten major divisions as in the preceding units. Each of these major divisions is in turn divided into two minor division divisions each having a value of one-half or .5 of the major division or .05 of unit 4. When we have a three digit number in this unit the third digit must be estimated unless it is a 5.

Examples: Number 435 is read unit 4 plus 3 major divisions plus 1 minor division; number 43716 is read unit 4 plus 3 major divisions plus 1 minor division which gives 435 and then add an estimated two-fifths of the next minor division giving 437. The remaining digits, 1 and 6, are too small to be considered so they are ignored as explained under the portion headed "THE SLIDE RULE." See Fig. 4 for these and other readings.

**FIG. 4 **

The remaining units on the scale are the same as unit 4 except that the division lines are closer together. Examples for units 5, 6, 7, 8 and 9 are shown in Fig. 5.

**FlG.5**

The number of 5.32 is read unit 5 plus 3 major divisions plus an estimated two-fifths of the next minor division; number .808 is read unit 8 plus 0 major divisions plus 1 minor division which gives 805 and then add three-fifths of the next minor division giving 808; number 9622 is read unit 9 plus 6 major divisions plus two-fifths of the next minor division. (The fourth digit 2 is omitted on the scale as previously explained.)

Multiplying 14 x 46 **FIG. 6**

As stated before, scales C and D are used in multiplication and are identical. Therefore. the divisions and reading of scale C will be the same as those found on scale D. The first line. number 1, at the left of the C scale is called the left index, while the last line at the right end of the, scale is called the right index.

When multiplying two numbers together we set either the right or the left index of the C scale directly over one of the numbers on the D scale; move the indicator so the hair line is over the number on the C scale; then read the answer under the hair line on the D scale.

Example 1: 3.4 x 7.2

Set the right index of the C scale over 7.2 on the D scale, move the indicator until the hair line is over 3.4 on the C scale. Read the 24.5 under the hair line on the D scale.

The most convenient way to locate the decimal point is to make a mental multiplication of the first digits in the factors. Then place the decimal point in the result so that the value is nearest that of the mental multiplication. Thus, 3 and 7, the first digits of our factors will give 21, indicating that the result 245 will become 24.5 instead of 2.45 or 245, as 24 is the nearest to our mental multiplication value of 21.

Example 2: 14 x 16 = 644

Set the left index of the C scale over 14 on the D scale, move the indicator so the hair line is over 46 on the C scale. Read the answer 644 under the hair line on the D scale. (See Fig. 6.)

From mental multiplication we see that 14, which is near 10, and 46, which is near 45, would give 10 x 45 or 450, showing that the answer would be in the hundreds, namely 644 instead of 64.4 or 6440.

Example 3: 18.72 x .356 = 6.66

Set the left index of the C scale over 1872 on the D scale, move the indicator until the hair line is over 356 on the C scale. Read the answer 666 under the hair line on the D scale. (See Fig 8.).

From observation we see that we are multiplying practically 20 x .3 giving 6. Therefore, our answer becomes 6.66 instead of 66.6 or .666.

Multiplication by the Integral Digit Method. (See page 21.)

As stated before scales C and D are used in multiplication and are identical. Therefore. the divisions and readings of scale C will be the same as those found on scale D. The first line, at number 1 at the left of the C scale, is called the left index, while the last line at the right end of the scale is called the right index.

When multiplying two numbers together we set either the right or the left index of the C scale directly over one of the numbers on the D scale; move the indicator so the hair line is over the other number on the C scale; then read the answer under the hair line on the D scale.

Example 1: 2 x 3 = 6

Set the slide so that the left index or number 1 of the C scale is directly over UNIT 2 on the D scale. Move the indicator so that the hair line is over UNIT 3 on the C scale, then read the answer 6 directly under the hair line on the D scale. (See Fig. 7.)

Example 2: 14 x 46 = 644

Set the slide so the left index of the C scale is over 14 on the D scale. Move the indicator so the hair line is over 46 on the C scale, then read the answer 644 under the hair line on the D scale. (See Fig.6.)

Multiplying 2 x 3 **FIG. 7**

It will be noted that the slide extended to the right in both of these operation and that the number of integral digits in each answer contained one less integral digit than the sum of the integral digits in both numbers. From this explanation we can establish a rule for multiplication when the slide extends to the right.

RULE: When the slide extends to the right, add the number of integral digits in the multiplier to those in the multiplicand and then subtract one; the result will be the number of integral digits in the answer.

Example 3: 18.72 x 0.356 = 6.66

Set the slide so that the left index of the C scale is over 187 on the D scale. Move the indicator so that the hair line is over 356 on the C scale, then read the answer 666 under the hair line on the U scale. (See Fig. 8.)

Multiplying 18.72 x 0.356 **FIG. 8**

The number 18.72 has plus 2 integral digits and 0.356 has zero integral digits; adding we get plus 2·integral digits. Since the slide extended to the right we subtract 1 leaving plus one integral digit, thus 666 becomes 6.66.

Example 4: 2351 x 41.2 = 96800

The left index of the C scale is set over 235 on the D scale; the hair line of the indicator over 412 on the C scale; read the number 968 under the hair line on the D scale. Number 2351 has 4 integral digits and number 41.2 has 2 integral digits; adding we obtain 6 integral digits. Since the slide moved to the right we subtract one leaving 5 integral digits in the answer. As we are only able to locate three numbers on the scale (three numbers are all that we can read in any answer), zeros must be added to complete the necessary number of integral digits. We will need two in this example making tire answer 96.800.

Problems:

183.2 x 27.42 Ans. 5020

31.52 x 19.7 Ans. 621

1.248 x 6.57 Ans. 8.20

257 x 0.313 Ans. 80.4

0.037 x 228 Ans. 8.44

233.42 x 0.00196 Ans. .457

It will be noted that the preceding problems in multiplication were such that the slide always extended to the right. We will now use numbers so that the slide will move to the left.

Example 1: 7 x 6 = 42

Set the right index of the C scale over 7 on the D scale, move the indicator so that the hair line is over 6 on the C scale, read the answer 42 under the hair line on the D scale. (See Fig 9.)

Multiplying 7 x 6 **FIG. 9**

Example 2: 8.3 x 27 = 224

Set the right index of the C scale over 83 on the D scale, move the indicator so that the hair line is over 27 on the C scale, read the answer 224 under the hair line on the D scale. (See Fig 10.)

Multiplying 8.3 x 27 **FIG. 10**

Example 3: 9.5 x 4.8 = 45.6

In the above example the slide extended to the left and the answer has as many integral digits as there were in the numbers themselves. Hence the rule: WHEN THE SLIDE EXTENDS TO THE LEFT, ADD THE NUMBER OF INTEGRAL DIGITS IN THE MULTIPLIER TO THOSE IN THE MULTIPLICAND AND THIS SUM WILL BE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER.

Example 4: 75.3 x 6.27 = 472

Number 75.3 has 2 integral digits and 6.27 has one integral digit, adding, we will have 3 integral digits in the answer: thus 472.

Example 5: 4387 x 0.0372 = 163

Number 4387 has 4 integral digits and 0.0372 has minus 1 integral digits, adding 4 to -1 we have 3 integral digits, thus the answer 163.

Example 6: 0.0057 x 0.0244 = .000139

Number 0.0057 has minus 2 (-2) integral digits and 0.0244 has minus 1 (-1) integral digit. Adding -2 to -1 we have -3 integral digits. Therefore, three zeros must be placed before number 138 making the answer 0.000138.

Problems:

75.21 x 485 Ans. 36.500

0.0672 x 3.48 Ans. 0.234

8.052 x 44.72 Ans. 0.360

9.122 x 0.1488 Ans. 1.357

When multiplying more than two numbers as 6 x 24 x 32, multiply 6 x 24 as has been explained; then take this answer and multiply it by 32 to obtain the final answer. Multiplying 6 x 24 = 144. 144 x 32 = 4610.

In the first operation, the slide extended to the left, therefore the answer will have 3 integral digits since there is 1 integral digit in number 6 and there are 2 integral digits in number 24. In the next operation, the slide extends to the right, so we must use the sum of the integral digits less one. Number 144 (answer of first operation) has 3 integral digits and number 32 has 2 integral digits. Adding we have 5 integral digits and less one leaves 4 integral digits in our final answer. As we were only able to read three numbers on the scale we must add 1 zero to make the necessary 4 integral digits.

Problems:

14 x 35 x 8.4 Ans. 4120.

3.76 x 0.786 x 185 Ans. 547.

27.5 x 12.5 x 0.27 Ans. 92.8

Location of Decimal Point in Division by the Mental Survey Method

In division we use the same scales, namely C and D, as were used in multiplication, only in reverse order.

When dividing, the hair line of the indicator is placed over the number to be "divided into" (numerator) on the D scale; the slide is moved to the right or left so that the number we are "dividing by" (denominator) is located on the C scale will fall directly under the hair line. The answer will be found under the index of the C scale on the D scale.

Example 1: 8 ÷ 4 = 2

Set the hair line of the indicator over 8 on the D scale, move the slide until 4 on the C scale is under the hair line; read the answer 2 under the left index on the D scale. (See Fig. 11.)

Dividing 8 by 4 **FIG. 11**

Example 2: 845 ÷ 25 = 33.8

From a mental survey we can see we have practically 800 ÷ 20, giving us two places in the answer; thus we obtain 33.8 instead of 338 or 3.38; or we can determine the decimal point in the following manner; 25 will go into 84 one digit and some left over; 25 will go into the left over amount and 5 another digit, thus giving 2 digits in our answer, so we have 33.8. The setting of the scale is shown in fig. 13.

Example 3:, 6187 ÷ 24.8 = 249

Here 24 will go into 61 one digit and give a remainder; this remainder and 8 can be divided by 24 making another digit in the answer and a remainder. Again this remainder and 7 can be divided by 24 making a third digit in the answer; thus we have three digits, 249 as the answer. Or we can say we have roughly 6000 ÷ 24, giving three numbers or digits in the answer.

In division we use the same scales as are used in multiplication, namely: the C and D scales because division is the reverse operation to multiplication.

When dividing two numbers the hair line of the indicator is placed over the numerator or dividend on the D scale; the slide is moved to the right or left so that the denominator or divisor located on tile C scale will fall directly under the hair line; read the answer under the index of the C scale on the D scale.

Example 1: 8 ÷ 4 = 2

Set the hair line of the indicator over 8 on the D scale; move the slide until 4 on the C scale is under the hair line; read the answer 2 on the D scale under the left index of the C scale. (See Fig. 11.)

Example 2: 48 ÷ 3 = 16

Set the hair line of the indicator over 48 on the D scale, move the slide until 3 of the C scale is under the hair line, read the answer 16 on D scale under the left index of the C scale. (See Fig. 12.)

Dividing 48 by 3 **FIG. 12**

Example 3: 845 ÷ 25 = 33.8

Set the hair line of the indicator over 845 on the D scale, move the slide until 25 of the C scale is under the hair line, read the answer 33.8 on the D scale under the left index of the C scale. (See Fig. 13.)

Dividing 845 by 25 **FIG. 13**

The user will note that in the preceding examples the slide moved to the left. When this occurs the rule for finding the number of integral digits in the answer is as follows:

IN DIVISION, WHEN THE SLIDE MOVES TO THE RIGHT, SUBTRACT THE NUMBER OF INTEGRAL DIGITS IN THE DIVISOR FROM THE NUMBER OF INTEGRAL DIGITS IN THE DIVIDEND AND THEN ADD 1. THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER.

Referring to example No.1, the dividend 8 has one integral digit and the divisor has one integral digit; subtracting the one integral digit in the dividend leaves zero, but since our slide moves to the right, we add one making 0 + 1 = 1. Therefore. our answer will have one integral digit.

Using example No.2, the dividend 48 has two integral digits and the divisor 3 has one integral digit. Subtracting one from two and then adding 1 we obtain two integral digits in the answer.

With example No.3, the dividend 845 has three integral digits and the divisor 25 has two integral digits. Subtracting two from three and then adding 1 we obtain two integral digits in the answer.

Problems:

72.6 ÷ 1.83 Ans. 39.7

6187 ÷ 24.8 Ans. 249

5.493 ÷ 37 Ans. 0.1484

813.6 ÷ 672 Ans. 1.21

Example 4: 763 ÷ .027 = 28,200

Set the hair line over 763 on the C scale, move the slide until 27 is under the hair line, read the number 282 on the D scale under the index of the C scale. The number 763 has three integral digits, and .027 has minus one integral digit. Since we are dividing, we subtract the integral digit of the divisor from those in the dividend as stated in the rule. But here we are subtracting a minus one from a plus three, which will give four as explained under "Subtraction of Integral Digits." Also the slide extended to the right, so we must add 1 to the result just obtained by subtracting the integral digits, making 4 + 1 or 5 integral digits in our answer. Therefore. our answer 28200; adding the two zeros to make the necessary five integral digits. (See Fig.14.)

Dividing 763 by 0.027 **FIG. 14**

Example 5: .0851 ÷ .0362 = 2.35

Set the hair line over 851 on the D scale, move the slide until 362 is under the hair line, read 235 on the D scale under the index. Here the dividend has a minus one integral digit and the divisor also has a minus one integral digit. Subtracting minus one from minus one we get zero, but the slide goes to the right so we must add one to zero making one integral digit in the answer. Therefore. the reading 235 becomes 2.35. (See Fig. 15.)

Dividing 0.0851 by 0.0362 **FIG. 15**

Example 6: 3852 ÷ 725 = 5.31

Example 7: 143.7 ÷ 9.36 = 15.35

In the examples above, 3852 has four integral digits while 725 described except that the slide will move to the left and the answer will be found under the right index of the C scale. When the slide moves to the left we have the following rule: IN DIVISION, WHEN THE SLIDE MOVES TO THE LEFT, SUBTRACT THE NUMBER OF INTEGRAL DIGITS IN THE DIVISOR FROM THE NUMBER OF INTEGRAL DIGITS IN THE DIVIDEND AND THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER.

In the example above, 3852 has four integral digits while 725 has three. Subtracting, we have one integral digit in the answer, giving 5.32. In example 7, 143.7 has three integral digits while 9.36 has one. Subtracting one from three gives two integral digits in the answer, so we have 15.36.

Example 8: 26.8 -÷ 0.0652 = 411

Here 26.8 the dividend, has two integral digits and .0652, the divisor, has minus one integral digit. Subtracting minus one from two we obtain three, which is the number of integral digits in the answer.

In this problem the rule of subtraction of integral digits is used, as previously explained.

Problems:

4873 ÷ 562 Ans. 8.67

93.14 ÷ 472 Ans. 0.197

0.0568 ÷ 0.027 Ans. 2.1

6.35 ÷ 18.2 Ans. 0.349

Scale A.

Scale A is divided into two halves, called the left and the right half. (See Fig. 1.)

The left half is divided into 9 unequal divisions, similar to scale D, and these divisions are also called UNIT divisions.

Unit 1 is divided just as unit 2 is on the D scale, so the divisions will have the same value as did unit 2 of the D scale. Units 2, 3 and 4 are like unit 4 of the D scale and they have the same value as did unit 4 of the D scale. Unit 5 of the A scale is divided in ten major divisions only, so the third digit of any number must be estimated. This is also true of units 6, 7, 8 and 9.

The right half of the A scale is divided as in the left half, therefore the values of the divisions will be the same.

As stated before, scales A and B are identical, as the readings on scale B will be the same as the reading on scale A.

Continue with Part 2

Posted 5/20/2013