[Transcriber's Notes] Conventional mathematical notation requires specialized fonts andtypesetting conventions. I have adopted modern computer programmingnotation using only ASCII characters. The square root of 9 is thusrendered as square_root(9) and the square of 9 is square(9). 10 divided by 5 is (10/5) and 10 multiplied by 5 is (10 * 5 ). The DOC file and TXT files otherwise closely approximate the originaltext. There are two versions of the HTML files, one closelyapproximating the original, and a second with images of the slide rulesettings for each example. By the time I finished engineering school in 1963, the slide rule was awell worn tool of my trade. I did not use an electronic calculator foranother ten years. Consider that my predecessors had little else touse--think Boulder Dam (with all its electrical, mechanical andconstruction calculations). Rather than dealing with elaborate rules for positioning the decimalpoint, I was taught to first "scale" the factors and deal with thedecimal position separately. For example: 1230 * . 000093 =1. 23E3 * 9. 3E-51. 23E3 means multiply 1. 23 by 10 to the power 3. 9. 3E-5 means multiply 9. 3 by 0. 1 to the power 5 or 10 to the power -5. The computation is thus1. 23 * 9. 3 * 1E3 * 1E-5The exponents are simply added. 1. 23 * 9. 3 * 1E-2 =11. 4 * 1E-2 =. 114 When taking roots, divide the exponent by the root. The square root of 1E6 is 1E3The cube root of 1E12 is 1E4. When taking powers, multiply the exponent by the power. The cube of 1E5 is 1E15. [End Transcriber's Notes] INSTRUCTIONSfor using aSLIDERULESAVE TIME!DO THE FOLLOWING INSTANTLY WITHOUT PAPER AND PENCILMULTIPLICATIONDIVISIONRECIPROCAL VALUESSQUARES & CUBESEXTRACTION OF SQUARE ROOTEXTRACTION OF CUBE ROOTDIAMETER OR AREA OF CIRCLE [Illustration: Two images of a slide rule. ] INSTRUCTIONS FOR USING A SLIDE RULE The slide rule is a device for easily and quickly multiplying, dividingand extracting square root and cube root. It will also perform anycombination of these processes. On this account, it is found extremelyuseful by students and teachers in schools and colleges, by engineers, architects, draftsmen, surveyors, chemists, and many others. Accountantsand clerks find it very helpful when approximate calculations must bemade rapidly. The operation of a slide rule is extremely easy, and it iswell worth while for anyone who is called upon to do much numericalcalculation to learn to use one. It is the purpose of this manual toexplain the operation in such a way that a person who has never beforeused a slide rule may teach himself to do so. DESCRIPTION OF SLIDE RULE The slide rule consists of three parts (see figure 1). B is the body ofthe rule and carries three scales marked A, D and K. S is the sliderwhich moves relative to the body and also carries three scales marked B, CI and C. R is the runner or indicator and is marked in the center witha hair-line. The scales A and B are identical and are used in problemsinvolving square root. Scales C and D are also identical and are usedfor multiplication and division. Scale K is for finding cube root. ScaleCI, or C-inverse, is like scale C except that it is laid off from rightto left instead of from left to right. It is useful in problemsinvolving reciprocals. MULTIPLICATION We will start with a very simple example: Example 1: 2 * 3 = 6 To prove this on the slide rule, move the slider so that the 1 at theleft-hand end of the C scale is directly over the large 2 on the D scale(see figure 1). Then move the runner till the hair-line is over 3 on theC scale. Read the answer, 6, on the D scale under the hair-line. Now, let us consider a more complicated example: Example 2: 2. 12 * 3. 16 = 6. 70 As before, set the 1 at the left-hand end of the C scale, which we willcall the left-hand index of the C scale, over 2. 12 on the D scale (Seefigure 2). The hair-line of the runner is now placed over 3. 16 on the Cscale and the answer, 6. 70, read on the D scale. METHOD OF MAKING SETTINGS In order to understand just why 2. 12 is set where it is (figure 2), notice that the interval from 2 to 3 is divided into 10 large or majordivisions, each of which is, of course, equal to one-tenth (0. 1) of theamount represented by the whole interval. The major divisions are inturn divided into 5 small or minor divisions, each of which is one-fifthor two-tenths (0. 2) of the major division, that is 0. 02 of thewhole interval. Therefore, the index is set above 2 + 1 major division + 1 minor division = 2 + 0. 1 + 0. 02 = 2. 12. In the same way we find 3. 16 on the C scale. While we are on thissubject, notice that in the interval from 1 to 2 the major divisions aremarked with the small figures 1 to 9 and the minor divisions are 0. 1 ofthe major divisions. In the intervals from 2 to 3 and 3 to 4 the minordivisions are 0. 2 of the major divisions, and for the rest of the D (orC) scale, the minor divisions are 0. 5 of the major divisions. Reading the setting from a slide rule is very much like readingmeasurements from a ruler. Imagine that the divisions between 2 and 3 onthe D scale (figure 2) are those of a ruler divided into tenths of afoot, and each tenth of a foot divided in 5 parts 0. 02 of a foot long. Then the distance from one on the left-hand end of the D scale (notshown in figure 2) to one on the left-hand end of the C scale would he2. 12 feet. Of course, a foot rule is divided into parts of uniformlength, while those on a slide rule get smaller toward the right-handend, but this example may help to give an idea of the method of makingand reading settings. Now consider another example. Example 3a: 2. 12 * 7. 35 = 15. 6 If we set the left-hand index of the C scale over 2. 12 as in the lastexample, we find that 7. 35 on the C scale falls out beyond the body ofthe rule. In a case like this, simply use the right-hand index of the Cscale. If we set this over 2. 12 on the D scale and move the runner to7. 35 on the C scale we read the result 15. 6 on the D scale under thehair-line. Now, the question immediately arises, why did we call the result 15. 6and not 1. 56? The answer is that the slide rule takes no account ofdecimal points. Thus, the settings would be identical for all of thefollowing products: Example 3:a-- 2. 12 * 7. 35 = 15. 6b-- 21. 2 * 7. 35 = 156. 0c-- 212 * 73. 5 = 15600. D-- 2. 12 * . 0735 = . 156e-- . 00212 * 735 = . 0156 The most convenient way to locate the decimal point is to make a mentalmultiplication using only the first digits in the given factors. Thenplace the decimal point in the slide rule result so that its value isnearest that of the mental multiplication. Thus, in example 3a above, wecan multiply 2 by 7 in our heads and see immediately that the decimalpoint must be placed in the slide rule result 156 so that it becomes15. 6 which is nearest to 14. In example 3b (20 * 7 = 140), so we mustplace the decimal point to give 156. The reader can readily verify theother examples in the same way. Since the product of a number by a second number is the same as theproduct of the second by the first, it makes no difference which of thetwo numbers is set first on the slide rule. Thus, an alternative way ofworking example 2 would be to set the left-hand index of the C scaleover 3. 16 on the D scale and move the runner to 2. 12 on the C scale andread the answer under the hair-line on the D scale. The A and B scales are made up of two identical halves each of which isvery similar to the C and D scales. Multiplication can also be carriedout on either half of the A and B scales exactly as it is done on the Cand D scales. However, since the A and B scales are only half as long asthe C and D scales, the accuracy is not as good. It is sometimesconvenient to multiply on the A and B scales in more complicatedproblems as we shall see later on. A group of examples follow which cover all the possible combination ofsettings which can arise in the multiplication of two numbers. Example4: 20 * 3 = 605: 85 * 2 = 1706: 45 * 35 = 15757: 151 * 42 = 63428: 6. 5 * 15 = 97. 59: . 34 * . 08 = . 027210: 75 * 26 = 195011: . 00054 * 1. 4 = . 00075612: 11. 1 * 2. 7 = 29. 9713: 1. 01 * 54 = 54. 514: 3. 14 * 25 = 78. 5 DIVISION Since multiplication and division are inverse processes, division on aslide rule is done by making the same settings as for multiplication, but in reverse order. Suppose we have the example: Example 15: (6. 70 / 2. 12) = 3. 16 Set indicator over the dividend 6. 70 on the D scale. Move the slideruntil the divisor 2. 12 on the C scale is under the hair-line. Then readthe result on the D scale under the left-hand index of the C scale. Asin multiplication, the decimal point must be placed by a separateprocess. Make all the digits except the first in both dividend anddivisor equal zero and mentally divide the resulting numbers. Place thedecimal point in the slide rule result so that it is nearest to themental result. In example 15, we mentally divide 6 by 2. Then we placethe decimal point in the slide rule result 316 so that it is 3. 16 whichis nearest to 3. A group of examples for practice in division follow: Example16: 34 / 2 = 1717: 49 / 7 = 718: 132 / 12 = 1119: 480 / 16 = 3020: 1. 05 / 35 = . 0321: 4. 32 / 12 = . 3622: 5. 23 / 6. 15 = . 8523: 17. 1 / 4. 5 = 3. 824: 1895 / 6. 06 = 31325: 45 / . 017 = 2647 THE CI SCALE If we divide one (1) by any number the answer is called the reciprocalof the number. Thus, one-half is the reciprocal of two, one-quarter isthe reciprocal of four. If we take any number, say 14, and multiply itby the reciprocal of another number, say 2, we get: Example 26: 14 * (1/2) = 7 which is the same as 14 divided by two. This process can be carried outdirectly on the slide rule by use of the CI scale. Numbers on the CIscale are reciprocals of those on the C scale. Thus we see that 2 on theCI scale comes directly over 0. 5 or 1/2 on the C scale. Similarly 4 onthe CI scale comes over 0. 25 or 1/4 on the C scale, and so on. To doexample 26 by use of the CI scale, proceed exactly as if you were goingto multiply in the usual manner except that you use the CI scale insteadof the C scale. First set the left-hand index of the C scale over 14 onthe D scale. Then move the indicator to 2 on the CI scale. Read theresult, 7, on the D scale under the hair-line. This is really anotherway of dividing. THE READER IS ADVISED TO WORK EXAMPLES 16 TO 25 OVERAGAIN BY USE OF THE CI SCALE. SQUARING AND SQUARE ROOT If we take a number and multiply it by itself we call the result thesquare of the number. The process is called squaring the number. If wefind the number which, when multiplied by itself is equal to a givennumber, the former number is called the square root of the given number. The process is called extracting the square root of the number. Boththese processes may be carried out on the A and D scales of a sliderule. For example: Example 27: 4 * 4 = square( 4 ) = 16 Set indicator over 4 on D scale. Read 16 on A scale under hair-line. Example 28: square( 25. 4 ) = 646. 0 The decimal point must be placed by mental survey. We know thatsquare( 25. 4 ) must be a little larger than square( 25 ) = 625so that it must be 646. 0. To extract a square root, we set the indicator over the number on the Ascale and read the result under the hair-line on the D scale. When weexamine the A scale we see that there are two places where any givennumber may be set, so we must have some way of deciding in a given casewhich half of the A scale to use. The rule is as follows: (a) If the number is greater than one. For an odd number of digits tothe left of the decimal point, use the left-hand half of the A scale. For an even number of digits to the left of the decimal point, use theright-hand half of the A scale. (b) If the number is less than one. For an odd number of zeros to theright of the decimal point before the first digit not a zero, use theleft-hand half of the A scale. For none or any even number of zeros tothe right of the decimal point before the first digit not a zero, usethe right-hand half of the A scale. Example 29: square_root( 157 ) = 12. 5 Since we have an odd number of digits set indicator over 157 onleft-hand half of A scale. Read 12. 5 on the D scale under hair-line. Tocheck the decimal point think of the perfect square nearest to 157. Itis 12 * 12 = 144, so that square_root(157) must be a little more than 12 or12. 5. Example 30: square_root( . 0037 ) = . 0608 In this number we have an even number of zeros to the right of thedecimal point, so we must set the indicator over 37 on the right-handhalf of the A scale. Read 608 under the hair-line on D scale. To placethe decimal point write: square_root( . 0037 ) = square_root( 37/10000 ) = 1/100 square_root( 37 ) The nearest perfect square to 37 is 6 * 6 = 36, so the answer should bea little more than 0. 06 or . 0608. All of what has been said about use ofthe A and D scales for squaring and extracting square root appliesequally well to the B and C scales since they are identical to the A andD scales respectively. A number of examples follow for squaring and the extraction of squareroot. Example31: square( 2 ) = 432: square( 15 ) = 22533: square( 26 ) = 67634: square( 19. 65 ) = 38635: square_root( 64 ) = 836: square_root( 6. 4 ) = 2. 5337: square_root( 498 ) = 22. 538: square_root( 2500 ) = 5039: square_root( . 16 ) = . 0440: square_root( . 03 ) = . 173 CUBING AND CUBE ROOT If we take a number and multiply it by itself, and then multiply theresult by the original number we get what is called the cube of theoriginal number. This process is called cubing the number. The reverseprocess of finding the number which, when multiplied by itself and thenby itself again, is equal to the given number, is called extracting thecube root of the given number. Thus, since 5 * 5 * 5 = 125, 125 is thecube of 5 and 5 is the cube root of 125. To find the cube of any number on the slide rule set the indicator overthe number on the D scale and read the answer on the K scale under thehair-line. To find the cube root of any number set the indicator overthe number on the K scale and read the answer on the D scale under thehair-line. Just as on the A scale, where there were two places where youcould set a given number, on the K scale there are three places where anumber may be set. To tell which of the three to use, we must make useof the following rule. (a) If the number is greater than one. For 1, 4, 7, 10, etc. , digits tothe left of the decimal point, use the left-hand third of the K scale. For 2, 5, 8, 11, etc. , digits to the left of the decimal point, use themiddle third of the K scale. For 3, 6, 9, 12, etc. , digits to the leftof the decimal point use the right-hand third of the K scale. (b) If the number is less than one. We now tell which scale to use bycounting the number of zeros to the right of the decimal point beforethe first digit not zero. If there are 2, 5, 8, 11, etc. , zeros, use theleft-hand third of the K scale. If there are 1, 4, 7, 10, etc. , zeros, then use the middle third of the K scale. If there are no zeros or 3, 6, 9, 12, etc. , zeros, then use the right-hand third of the K scale. Forexample: Example 41: cube_root( 185 ) = 5. 70 Since there are 3 digits in the given number, we set the indicator on185 in the right-hand third of the K scale, and read the result 570 onthe D scale. We can place the decimal point by thinking of the nearestperfect cube, which is 125. Therefore, the decimal point must be placedso as to give 5. 70, which is nearest to 5, the cube root of 125. Example 42: cube_root( . 034 ) = . 324 Since there is one zero between the decimal point and the first digitnot zero, we must set the indicator over 34 on the middle third of the Kscale. We read the result 324 on the D scale. The decimal point may beplaced as follows: cube_root( . 034 ) = cube_root( 34/1000 ) = 1/10 cube_root( 34 ) The nearest perfect cube to 34 is 27, so our answer must be close toone-tenth of the cube root of 27 or nearly 0. 3. Therefore, we must placethe decimal point to give 0. 324. A group of examples for practice inextraction of cube root follows: Example43: cube_root( 64 ) = 444: cube_root( 8 ) = 245: cube_root( 343 ) = 746: cube_root( . 000715 ) = . 089447: cube_root( . 00715 ) = . 19348: cube_root( . 0715 ) = . 41549: cube_root( . 516 ) = . 80350: cube_root( 27. 8 ) = 3. 0351: cube_root( 5. 49 ) = 1. 7652: cube_root( 87. 1 ) = 4. 43 THE 1. 5 AND 2/3 POWER If the indicator is set over a given number on the A scale, the numberunder the hair-line on the K scale is the 1. 5 power of the givennumber. If the indicator is set over a given number on the K scale, thenumber under the hair-line on the A scale is the 2/3 power of the givennumber. COMBINATIONS OF PROCESSES A slide rule is especially useful where some combination of processes isnecessary, like multiplying 3 numbers together and dividing by a third. Operations of this sort may be performed in such a way that the finalanswer is obtained immediately without finding intermediate results. 1. Multiplying several numbers together. For example, suppose it isdesired to multiply 4 * 8 * 6. Place the right-hand index of the C scaleover 4 on the D scale and set the indicator over 8 on the C scale. Now, leaving the indicator where it is, move the slider till the right-handindex is under the hairline. Now, leaving the slider where it is, movethe indicator until it is over 6 on the C scale, and read the result, 192, on the D scale. This may be continued indefinitely, and so as manynumbers as desired may be multiplied together. Example 53: 2. 32 * 154 * . 0375 * . 56 = 7. 54 2. Multiplication and division. Suppose we wish to do the following example: Example 54: (4 * 15) / 2. 5 = 24 First divide 4 by 2. 5. Set indicator over 4 on the D scale and move theslider until 2. 5 is under the hair-line. The result of this division, 1. 6, appears under the left-hand index of the C scale. We do not need towrite it down, however, but we can immediately move the indicator to 15on the C scale and read the final result 24 on the D scale under thehair-line. Let us consider a more complicated problem of the same type: Example 55: (30/7. 5) * (2/4) * (4. 5/5) * (1. 5/3) = . 9 First set indicator over 30 on the D scale and move slider until 7. 5 onthe C scale comes under the hairline. The intermediate result, 4, appears under the right-hand index of the C scale. We do not need towrite it down but merely note it by moving the indicator until thehair-line is over the right-hand index of the C scale. Now we want tomultiply this result by 2, the next factor in the numerator. Since twois out beyond the body of the rule, transfer the slider till the other(left-hand) index of the C scale is under the hair-line, and then movethe indicator to 2 on the C scale. Thus, successive division andmultiplication is continued until all the factors have been used. Theorder in which the factors are taken does not affect the result. With alittle practice you will learn to take them in the order which willrequire the fewest settings. The following examples are for practice: Example 56: (6/3. 5) * (4/5) * (3. 5/2. 4) * (2. 8/7) = . 8 Example 57: 352 * (273/254) * (760/768) = 374 An alternative method of doing these examples is to proceed exactly asthough you were multiplying all the factors together, except thatwhenever you come to a number in the denominator you use the CI scaleinstead of the C scale. The reader is advised to practice both methodsand use whichever one he likes best. 3. The area of a circle. The area of a circle is found by multiplying3. 1416=PI by the square of the radius or by one-quarter the square ofthe diameter Formula: A = PI * square( R ) A = PI * ( square( D ) / 4 ) Example 58: The radius of a circle is 0. 25 inches; find its area. Area = PI * square(0. 25) = 0. 196 square inches. Set left-hand index of C scale over 0. 25 on D scale. Square(0. 25) nowappears above the left-hand index of the B scale. This can be multipliedby PI by moving the indicator to PI on the B scale and reading theanswer . 196 on the A scale. This is an example where it is convenient tomultiply with the A and B scales. Example 59: The diameter of a circle is 8. 1 feet. What is its area? Area = (PI / 4) * square(8. 1) = . 7854 * square(8. 1) = 51. 7 sq. Inches. Set right-hand index of the C scale over 8. 1 on the D scale. Move theindicator till hair-line is over . 7854 (the special long mark near 8) atthe right hand of the B scale. Read the answer under the hair-line onthe A scale. Another way of finding the area of a circle is to set 7854on the B scale to one of the indices of the A scale, and read the areafrom the B scale directly above the given diameter on the D scale. 4. The circumference of a circle. Set the index of the B scale to thediameter and read the answer on the A scale opposite PI on the B scale Formula: C = PI * D C = 2 * PI * R Example 60: The diameter of a circle is 1. 54 inches, what is itscircumference? Set the left-hand index of the B scale to 1. 54 on the A scale. Read thecircumference 4. 85 inches above PI on the B scale. EXAMPLES FOR PRACTICE 61: What is the area of a circle 32-1/2 inches in diameter?Answer 830 sq. Inches 62: What is the area of a circle 24 inches in diameter?Answer 452 sq. Inches 63: What is the circumference of a circle whose diameter is 95 feet?Answer 298 ft. 64: What is the circumference of a circle whose diameter is 3. 65 inches?Answer 11. 5 inches 5. Ratio and Proportion. Example 65: 3 : 7 : : 4 : Xor(3/7) = (4/x)Find X Set 3 on C scale over 7 on D scale. Read X on D scale under 4 on Cscale. In fact, any number on the C scale is to the number directlyunder it on the D scale as 3 is to 7. PRACTICAL PROBLEMS SOLVED BY SLIDE RULE 1. Discount. A firm buys a typewriter with a list price of $150, subject to adiscount of 20% and 10%. How much does it pay? A discount of 20% means 0. 8 of the list price, and 10% more means0. 8 X 0. 9 X 150 = 108. To do this on the slide rule, put the index of the C scale opposite 8 onthe D scale and move the indicator to 9 on the C scale. Then move theslider till the right-hand index of the C scale is under the hairline. Now, move the indicator to 150 on the C scale and read the answer $108on the D scale. Notice that in this, as in many practical problems, there is no question about where the decimal point should go. 2. Sales Tax. A man buys an article worth $12 and he must pay a sales tax of 1. 5%. Howmuch does he pay? A tax of 1. 5% means he must pay 1. 015 * 12. 00. Set index of C scale at 1. 015 on D scale. Move indicator to 12 on Cscale and read the answer $12. 18 on the D scale. A longer but more accurate way is to multiply 12 * . 015 and add theresult to $12. 3. Unit Price. A motorist buys 17 gallons of gas at 19. 5 cents per gallon. How muchdoes he pay? Set index of C scale at 17 on D scale and move indicator to 19. 5 on Cscale and read the answer $3. 32 on the D scale. 4. Gasoline Mileage. An automobile goes 175 miles on 12 gallons of gas. What is the averagegasoline consumption? Set indicator over 175 on D scale and move slider till 12 is underhair-line. Read the answer 14. 6 miles per gallon on the D scale underthe left-hand index of the C scale. 5. Average Speed. A motorist makes a trip of 256 miles in 7. 5 hours. What is his averagespeed? Set indicator over 256 on D scale. Move slider till 7. 5 on the C scaleis under the hair-line. Read the answer 34. 2 miles per hour under theright-hand index of the C scale. 6. Decimal Parts of an Inch. What is 5/16 of an inch expressed as decimal fraction? Set 16 on C scale over 5 on D scale and read the result . 3125 inches onthe D scale under the left-hand index of the C scale. 7. Physics. A certain quantity of gas occupies 1200 cubic centimeters at atemperature of 15 degrees C and 740 millimeters pressure. What volumedoes it occupy at 0 degrees C and 760 millimeters pressure? Volume = 1200 X (740/760) * (273/288) = 1100 cubic cm. Set 760 on C scale over 12 on D scale. Move indicator to 740 on C scale. Move slider till 288 on C scale is under hair-line. Move indicator to273 on C scale. Read answer, 1110, under hair-line on D scale. 8. Chemistry. How many grams of hydrogen are formed when 80 grams of zinc react withsufficient hydrochloric acid to dissolve the metal? (80 / X ) = ( 65. 4 / 2. 01) Set 65. 4 on C scale over 2. 01 on D scale. Read X = 2. 46 grams under 80 on C scale. In conclusion, we want to impress upon those to whom the slide rule is anew method of doing their mathematical calculations, and also theexperienced operator of a slide rule, that if they will form a habit of, and apply themselves to, using a slide rule at work, study, or duringrecreations, they will be well rewarded in the saving of time andenergy. ALWAYS HAVE YOUR SLIDE RULE AND INSTRUCTION BOOK WITH YOU, thesame as you would a fountain pen or pencil. The present day wonders of the twentieth century prove that there is noend to what an individual can accomplish--the same applies to the sliderule. You will find after practice that you will be able to do manyspecialized problems that are not outlined in this instruction book. Itdepends entirely upon your ability to do what we advocate and to beslide-rule conscious in all your mathematical problems. CONVERSION FACTORS 1. Length 1 mile = 5280 feet =1760 yards 1 inch = 2. 54 centimeters 1 meter = 39. 37 inches 2. Weight (or Mass) 1 pound = 16 ounces = 0. 4536 kilograms 1 kilogram = 2. 2 pounds 1 long ton = 2240 pounds 1 short ton=2000 pounds 3. Volume 1 liquid quart = 0. 945 litres 1 litre = 1. 06 liquid quarts 1 U. S. Gallon = 4 quarts = 231 cubic inches 4. Angular Measure 3. 14 radians = PI radians = 180 degrees 1 radian = 57. 30 degrees 5. Pressure 760 millimeters of mercury = 14. 7 pounds per square inch 6. Power 1 horse power = 550 foot pounds per second = 746 watts 7. Miscellaneous 60 miles per hour = 88 feet per second 980 centimeters per second per second= 32. 2 feet per second per second= acceleration of gravity. 1 cubic foot of water weighs 62. 4 pounds 1 gallon of water weighs 8. 34 pounds Printed in U. S. A. INSTRUCTIONS FOR USING A SLIDE RULECOPYRIGHTED BY W. STANLEY & CO. Commercial Trust Building, Philadelphia, Pa.