BEGINNERS' SLIDE RULE MANUAL SCIENCE By Otto G. Brown and H. Grady Rylander University Interscholastic league BEGINNERS' SLIDE RULE MANUAL BY OTTO G. BROWN Instructor of Mechanical Engineering and State Slide Rule Director The University of Texas AND H . GRADY RYLANDER Associate Professor of Mechanical Engineering The University of Texas The benefits of education and of useful knowledge, generally diffused through a community, are essential to the preserva­tion of a free government. SAM HOUSTON while guided and controlled by virtue, the noblest attribute of man. It is the only dictator that freemen acknowledge, and the only security which freemen desire. MIRABEAU B. LAMAR THE UNIVERSITY OF TEXAS PUBLICATION NUMBER 7005 MARCH I, 1970 PUBLISHED TWICE A MONTH BY THE UNIVERSITY OF TEXAS, UNIVERSITY STATION, AUSTIN, TEXAS, 78712, SECOND-CLASS POSTAGE PAID AT AUSTIN, TEXAS. Table of Contents ., I. Introduction I II. Selecting a Slide Rule 8 III. Parts of a Slide Rule 10 IV. Care of the Rule 11 v. Adjusting a Slide Rule 12 VI. Common Scales of the Slide Rule 15 VII. Reading a Scale 16 VIII. Multiplication 19 IX. Division 28 x. Placement of the Decimal Point 31 XI. Complex Multiplication and Division 34 XII. Squares and Square Roots 38 XIII. Cubes and Cube Roots . 41 XIV. Inverted Scales . 46 xv. Conclusion 47 XVI. Practice Problems 48 XVII. Answers 52 Acknowledgments The authors wish to express sincere appreciation to the following slide rule manufacturers for the use of their slide rules in preparing the illustrations in this manual. EuaENE DrnTZGEN CoMPANY. KEUFFEL & ESSER COMPANY. PICKETT & ECKEL, INC. FR&DERICK PosT CoMPANY. Statement on Equal Educational Opportunity With respect to the admission and education of students, with respect to the availability of student loans, grants, scholarships, and job opportunities, with re· spect to the employment and promotion of teaching and nonteaching personnel, with respect to the student and faculty activities conducted on premises owned or occupied by the University, and with respect to student and faculty housing situ· ated on premises owned or occupied by the University, The University of Texas at Austin shall not discriminate either in favor of or against any person on ac­count of his or her race, creed, color, or national origin. Preface This revision of the University Interscholastic League Slide Rule Manual has been written for the express purpose of helping the begin­ner to understand more clearly the basic slide rule operations. In particular, the fundamental operations and the location of the deci­mal point in these operations have been completely rewritten while other topics are partially revised. Also, the expanded illustrations should be more helpful in reading the settings in the examples. These illustrations are shown for the most generally used slide rule, the ten inch straight rule. Also, at the beginning of the answer section in this manual, some methods of answer presentation are given which are the basis for the judges' scoring in formal competition. You will discover that advanced slide rule operations are not dis­cussed in this manual. These operations, as well as some of the more complicated but fundamental operations, can best be mastered by consulting your coach or instructor. The specific rules for University Interscholastic League Slide Rule Competition are not included in this manual. However, for this in­formation, the reader is referred to "Constitution and Contest Rules of the University Interscholastic League." With the ever-growing popularity of the slide rule in a great many fields, not only are you preparing for interesting individual or club competition, but you are also paving the way for the future by learn­ing its proficient use. RooNEY J. Kmo, Director University Interscholastic League I. Introduction The slide rule is a very handy instrument that serves as a medium for performing rapid calculations. Due to its relatively small size, it is readily portable and may be easily carried about in one's pocket or stored in a desk. There are countless numbers and types of problems that it can aid in solving. It can multiply, divide, give ratios and pro­portions, raise numbers to powers, take roots of numbers, give the trigonometric and hyperbolic functions of angles, give the logarithms of numbers to any base, indicate the reciprocals of numbers, give the areas of circles, solve certain types of equations directly, and per­form many other useful operations at a speed many times greater than that of longhand calculations. It can even be induced to add and subtract. The first slide rule appeared in 1620 as a result of Gunter's inven­tion of the straight logarithmic scale based upon Napier's invention of logarithms. Even though it did not look like our present-day rules, its principle of operation was the same as that employed by the most modern, complicated slide rules now in existence. This early an­cestor of our slide rule consisted of a single scale on which the settings were made by employing dividers to indicate line segments the lengths of which were logarithmically proportional to the magnitudes of numbers. It was rather simple and crude, but it performed its job well. In the centuries that have followed, many different types of slide rules have appeared. Some called rectilinear rules are straight; others in the form of a disk are called circular rules ; a few have appeared in the form of a cylinder consisting of a number of rotating drums. Other special forms of rules, particularly useful for performing spe­cialized tasks required by engineering, science, industry, or the mili­tary forces, have been developed. A few of the many names that have been given to these different types of rules are: polyphase, log log duplex decitrig, log log vector, binary, deci log log, vector hyperbolic, flight calculator, power computer, versalog, and other equally queer sounding names to the beginning slide rule operator. Probably the most commonly used rule is the rectilinear rule, al­though the circular rules are used by a large number of individuals. 7 For this reason, this manual will employ the straight slide rule for its instructional examples. This publication is designed to assist any interested individual to become an efficient operator or a slide rule. With adequate practice he should become a proficient operator. The beginner should keep his rule before him while he is reading and studying these instruc­tions. He should also make all settings indicated in the illustrative examples, and should compute the answers for all the exercises a num­ber of times until he has assured himself that he has mastered that particular phase of calculations. The principles involved are rather simple and should be understood with a small amount of study. How­ever, a great amount of practice is absolutely necessary before one gains complete confidence in his ability to operate the rule profici­ently. Learning to use a slide rule is like learning algebra; it cannot be done overnight or in a week's time. Consequently, the beginner, justifiably, should not be too disappointed over his seemingly slow progress in learning the efficient use of the rule. Practice exercises are given in this manual with each particular phase of instructions pertaining to that particular type of slide rule operation. Also, an additional number of practice problems of a more complex nature are listed in section XVI of this bulletin. Finally, the answers to all practice exercises and to the practice problems are given at the end (section XVII) to enable the operator to check his calculations. With the answers thus shown separately from the prob­lems, it permits one to solve the particular problem in its entirety be­fore checking the answer. One must remember that, although the slide rule is capable of do­ing a great many useful things when in the hand of a proficient ope­rator, it cannot think. It can never be more accurate than the readings or settings made on it by its user. This is the operator's particular task: think for the rule, and let it perform the mechanics of the operation; only then will you begin to reap the benefits of its use. II. Selecting a Slide Rule Any reputable manufacturer or dealer will furnish a price list of the various type of rules. At the end of this section some of the more reputable firms that produce slide rules are listed for information. The prices for the various rules will vary, depending upon the manu­ 8 facturer and the type of rule. Also, if a case is purchased, of course, the total cost will be more. Generally, however, it might be said that a first class ten inch rule might be purchased for a price in the range of twenty dollars. It is worthwhile to consider the purchase of a case for the rule, as it affords an excellent means of protection against most types of damage. Section V gives useful information on ad­justing a slide rule. In calculations where extreme accuracy is required, the twenty inch straight rule is probably the best choice for use, as it is capable of giving an accuracy of 1 part in 2000. However, for almost all cal­culations necessary in engineering, science, and related fields, the ten inch rule gives sufficient accuracy, 1 part in 1000. Another factor to be considered here is that, although the twenty inch straight rule is twice as accurate as the ten inch straight rule, it is not as handy to carry or store. In comparing straight and circular slide rules, it must be borne in mind that each has favorable factors. An eight to ten inch diameter circular rule gives about the same accuracy as a twenty inch straight rule. However, the time required for a particular operation is general­ly greater for the circular rule. On the other hand, the operator does not have to decide which index to use in order to prevent running out of range of the scale when using a circular rule. Slide rules are constructed of wood, metal, or plastic. Generally speaking, a plastic rule is cheaper in cost than either a metallic or a wooden rule. However, the plastic rule also has a greater tendency to warp with age and to chip when dropped. Wooden rules are af­fected by high humidity conditions in that the slide sometimes has a tendency to stick. Nevertheless, with care this condition can usually be avoided. Metallic rules generally remain in adjustment the best. However, many operators prefer the ease of reading the scales of the wooden rules rather than the lithographed or printed scales generally found on metallic rules. Probably the most satisfactory slide rule for all around use is the ten inch wooden or metallic slide rule. An accompanying case also serves as insurance against damage when not in use. The following manufacturers are among those who produce slide rules of excellent quality: Eugene Dietzgen Company, 318 Camp St., New Orleans, Louis­ iana 70130 9 Keuffel & Esser Company, 1701 Walker Ave., Houston, Texas 77001 Gilson Slide Rule Company, P. 0. Box 111, Stuart, Florida John Henschel Company, 195 Mairine St., Farmingdale, New York 11735 Pickett Inc., 436 Gutierrez St., Santa Barbara, California 93102 Frederick Post Company, Box 138, Houston, Texas 77002 F. Weber Company, 2000 Windrim, Philadelphia, Pennsylvania 19144 III. Parts of a Slide Rule The straight slide rule consists essentially of three parts; the body, the slide, and the cursor. These are as indicated in Figure 1. I TOP BODY MEMBER CURSOR SLIDE The main part of the slide rule is the body and consists of two outer bars held apart on each end by metallic plates. Both the front and re­verse faces of the body are covered with plastic coating or painted, and have lithographed, printed, or stamped scales. The body some­times is referred to as the stock. The second part of the rule is the slide, a long sliding member which moves between the two bars of the body. The slide, like the body, has scales on both its faces. This part is also called the slider. The cursor is the glass or plastic runner, generally framed in a spring loaded metallic holder, with a black or red hairline placed perpendicular to the direction of sliding of the cursor along the.body. Generally, a glass or plastic runner is provided for both the front and rear faces of the rule. The accompanying hairlines move together IO since they are carried by the same cursor. Sometimes the cursor is re­ferred to as the indicator. The horizontal rows of calibration marks on both the body and the slide are called scales. Each scale is named by a letter or letters gen­erally placed at both ends. It is interesting to note that, even though the most commonly used straight slide rule is referred to as a ten inch rule, its scales generally are twenty-five centimeters in length, approximately 0.16 inch less than ten inches. A few of the recent slide rules have scales that actually measure ten inches in length. IV. Care of the Rule One must remember that the slide rule is a precision instrument and measures should be taken to prevent damage to it which might affect its use. A slide rule case can do much to help prevent damage to the rule. The cases are usually made of leather but some are also made of plas­tic. A slide rule case can help prevent such damage as scratched scales, broken indicators, and warped slides or body members. The rule should not be stored where it is subjected to conditions of high temperature or high humidity, such as by an open window, in direct sunlight, or near a source of heat. This practice can, of course, cause warping and discoloring. Although, in some instances, chemicals are used in an effort to ease the motion of the slide, this practice should be avoided. The best so­lution for a sticky rule is proper adjustment. However, a slight appli­cation of powder will do much to alleviate the friction between slide and body. The use of an artgum eraser or carbon tetrachloride cleaner is recommended for use on discolored and dirty scales. However, care should be exerted in order to prevent damage to the figures on the scales. The use of soap and water is not recommended with wooden rules. The best way to clean the indicator is to disassemble it, clean all parts, reassemble it, and finally adjust it. The indicator can be cleaned somewhat by placing a strip of soft paper between the indicator and the body and, while applying a moderate force on the indicator, mov­ing it to and fro across the paper. The use of a bleaching agent, ammonia, alcohol, dye solvent, or naptha is useful in removing stains from the rule. However, care I I should be exerted in order to prevent damage to the calibration marks on the scale. Frequently these solutions will dissolve the dye or ink used to mark the scales or will even dissolve the material from which the scales are made. It is alarming how many slide rules are damaged by cigarettes. Careless smoking and slide rule operating are not winnin~ combi­nations. Above all things, handle the rule carefully. Exert caution not to drop it nor to step on it accidentally. After each cleaning of the rule, it should be checked and corrective adjustments should be made, if necessary. V. Adjusting a Slide Rule When purchasing a slide rule, one should check the serial number (Fig. 2) listed on the slide and the body of the rule to be sure that SERIAL NUMBERS they correspond. This precaution is necessary because it is possible, if one obtains a rule with a slide and body that have different serial numbers, that the two members will not operate as smoothly to­gether or may not give accurate results. This check need be made only in the case of straight slide rules. Since circular slide rules do not have a distinct slide separate from the body, this serial number check is not necessary. The first step necessa1y in adjusting a slide rule is to be sure that the slide is properly set in the body of the rule. That is, be sure that the front face of the slide is on the front face of the rule. The scales generally appearing on the front side of the slide are the CF, CIF, Cl, and C scales. The front face of the body generally contains the L, LL/, DF, D, LL2, LL3, and other scales. The next step is to align the left indexes of both the C and D scales. With this alignment, the right indexes of the same C and D scales must coincide. If these indexes do not coincide, the scales are not of the same length and the results obtained using these scales will be erratic. This particular trouble cannot be corrected. Thus, this align­ment should be checked when purchase is made. In addition, of course, the corresponding set of mating indexes on the reverse side should be checked to be sure they are of the same length. With the slide in the position where the above four pairs of indexes coincide, alignment of the top member of the body should be checked. The left and right indexes of the DF and CF scales on the front face of the rule should coincide. If they do not, loosen the body adjusting screws A and B (Fig. 3), move the top body member back and forth E F Fig. 3 . Slide Rule in Correct Adiustment. until these two pairs of indexes coincide, and tighten securely the ad­justing screws. When this adjustment has been made, turn the rule over without moving the slide relative to the body and note the alignment of the corresponding indexes on the reverse side. If these indexes on the reverse side are not in proper alignment, the rule cannot be corrected and, thus, this alignment also should be checked at time of purchase. With the slide set and the rule adjusted so the eight pairs of indexes coincide (four on front side and four on reYerse side), the indicator hairline now must be adjusted. To do this, loosen the indicator ad­justment screws ( C, D, E, and F, in Fig. 3) on on both sides of the rule, adjust the indicator until the hairline coincides with the \·ertical pairs of indexes on the left end of the front face of the rule, and then tighten the four adjusting screws. This same procedure should then be repeated on the reverse side of the rule in order to insure proper hairline alignment there also. Extreme care should be taken when per­forming these adjustments in order to prevent damage to the heads and threads of the adjusting screws. Figure 3 illustrates a slide rule in proper adjustment on the front side. The rule should also be tested for tightness. To·do this, mO\·e the slide back and forth between the two extreme positions of sliding. If the slide appears to move with great difficulty, or, on the other hand, moves too freely, the tightness of the rule should be adjusted. If the slide moves too stubbornly, the two body adjusting screws A and B (Fig. 3) should be loosened and the top body member should be moved away from the slide. The screws should be tightened and the slide can then be tested for freedom of movement. A trial and error method of adjustment should finally permit satisfactory operation. On the other hand, if the slide is adjusted too loosely (the slide falls under the effects of gravity only when the rule is held in a Yertical position), correct tightness can be effected in the following manner: loosen the two body adjusting screws (A and Bin Fig. 3), insert two pieces of thin paper (tracing paper or tissue paper) between the slide and the top body member as indicated in Figure 4, press each end of the top body member tightly to the slide, and tighten the body ad­justing screws. The slide should then be removed from the body, the traces of paper extracted, and the slide reinserted in the body. When making these adjustments on slide tightness, one must be careful not to disturb the alignment of the eight pairs of coincident indexes. Best results are usually obtained by loosening the screws at only one end of the rule at a time. The slide rule is now in correct adjustment and ready for efficient operation. In some cases, rules have warped slides or body members. The above adjustments cannot completely correct this condition, and if the movement of the slide is extremely jerky, the rule should be re­jected. VI. Common Scales of the Slide Rule Except for the same model rule made by the same manufacturer, it is very uncommon that two slide rules will happen to have the same number and kind of scales on them. Moreover, the placement of the various scales will vary from slide rule to slide rule. Therefore, the listing of the scales that follow is a more general listing according to their most generally known names. Also, the most important uses of these scales are given. C and D scales: Used for multiplication and division, or a combi­nation of multiplication and division, and in conjunction with A, B, and K scales for square and cube roots. CF and DF scales: Folded at 7T; otherwise identical to the C and D scales. The CF and DF scales, when used in conjunction with the C and D scales, permit direct calculations involving multiplication and division by -rr. In most cases where the wrong index has been used when multiplying on the C and D scales, the answer may be found on either the CF or DF scale. A and B scales: Used in conjunction with the D and C scales to obtain the squares and square roots of numbers. Also used for multi­plication by setting left index of B in first range of A or by setting right index of Bin second range of A. K scale: Used in conjunction with the C or D scale to obtain the cubes and cube roots of numbers. CI scale: An inverted C scale giving reciprocals of numbers di­rectly opposite on the C scale. DI scale: An inverted D scale giving reciprocals of numbers ap­pearing on the D scale. CIF scale: An inverted CF scale giving reciprocals of numbers ap­pearing on the CF scale. L scale: A uniformly divided scale that is used in conjunction with the D scale to give the mantissas of logarithms to the base ten. TRIG scales (T, ST, S) : Used in conjunction with the C or D scale to obtain the natural trigonometric functions of angles. The above scales are the most commonly found scales on slide rules; however, there are a number of other scales, some quite un­common, found on various slide rules. VII. Reading a Scale Nearly everyone at one time or another has had experience m measuring distances by use of a rule, yardstick, or tape calibrated in inches. A slide rule is similar to a straight twelve inch rule in that the numbers appearing on its scales are a measure of the magnitude of numbers rather than the magnitude of distances. On a straight 12­inch rule, the numbers appearing on the calibration indicate inches; however, between these there are several unnumbered lines indicating fractions of an inch. Thus, on the 12-inch straight rule shown (Fig. 5); the point at A designates a reading of 1%"or 1.625" on a decimal 1 5.751 8. 501 1 ''l'"l"'I''' '''l'''I 'I''I''' ··1· 1·! ~ 6 Fig. 5. Reading a linear Scale . 16 basis. Like a straight rule, the figures on a slide rule scale indicate numbers, and the division lines between figures indicate magnitude of numbers between the printed figures on the scales. On the slide rule, the scale divisions are not uniform. This is due to the fact that the slide rule operates on a logarithmic background. Nevertheless, the method of reading intermediate values is the same as for the straight 12-inch rule. Thus (Fig. 6) the point at A designates a reading on K I. I' 1. 1. 1. l.1.i.i.i.!.i.i.i.1.1.1,1.i.1.l .... 1....f....1....l....1....t....:....1.. J ....I. I. 1.1.1.1.1.1.1.1.\ : 1111111111111111111111111111111111111111111111111[111111111111111111111~11111111111111~11111111111111111[11111111~1111111~11 K I. 1.1.1.1.l.1.i.i.i.!.i.i.1 .1.l.1.1.1.i.l .... 1....f....1 .. ..l....1....t....f....l....L.l .1.1.1.1.t.1 .1.1.1.\ Cl ·1.1.i.1.1.t,1.1,1,i.!.1.1.1.1.t.i.1.1.i.l.1.1.1.1.l.1.1.1.1'.l.1,1,1.1.l.1.1,1.1'.!.1.1.11.1. l.1.1.1 ":l.1.'. i; I. I .1 .1. I. I :t....1....1.... 1 c L &111111n1111111m1mirm111111rn•1•111r11.T"'l"''l'"'l"''l'"'j""l"''l""l'"'l"n~m "'l'"'l'"'l""l""l"''l'"'l'"'l""f"l""l'"'l"''l'"'l""1'"'1'"'1'"'1'"1""1"''1"''1' @-1625 Fig. 6. Reading a Slide Rule Scale. the D slide rule scale of 1.625. However, as we shall see, this point could also represent a reading of 0.01625, 0.1625, 1.625, 16.25, 162.5, 16,250, or the same ordered combination of numbers ( 1625) with any number of zeros immediately following the combination 1625 and before the decimal point or any number of zeros immediately preced­ing the combination 1625 and after the decimal point. As odd as it may appear to the beginner, with few exceptions, the decimal point has nothing to do with the location of a number on the scales of a slide rule; it is the sequence of digits that determines this location. On either the C or D scales, the most generally used scales, the location of 0.002, 0.020, 0.200, 2.00, 20.0, 200.0, 2000.0, or 2 with any number of zeros either before or after it will fall at exactly the same position on the scale. Thus (Fig. 7) point A represents a scale reading of 167, point B represents a reading of 255, and point C represents a reading of 670. Note that decimal points have not been imposed on the above readings since "a slide rule setting is independent of the decimal point." You will notice that, in many cases, interpolation will be necessary to set a particular number on the C or D scale. Suppose, for example, 17 @167 ©255 LLt:J.m Lll ~:: If 111 1 cr ~111,1~Jni~1;1.1,1i1i1,111!~l,1~.,11 I·,11\"'M:\:,,,!•/1\'I!!•/ 1\ 1\i/1\•\1\·ii/i~!1111!\111lj1\~l11i1~·/1WM·\'1'•l•'1!I· 1l/,11,1f11i11,\WMl1~\' cir l""l"~'i'"'l'"''""l'"''""I' .. ·, .. ••1· •••1•···1 ••·; f"r:· '"'1·~·11""1··;1r·"1";~1""t";;r·"•":4l""'"~'J"" ··:11····1· •·:11····1 L Cl c 1 ~ • c c c l"1"1•11t'11"""'il•1"""':""'\"i1'1'"''1"\01"'11"i\"11l'1•111."1"\ 1\1.t1•1111"1i.11•\i.111\111.l1 "'1t"'11'\'111"il"''011'l"1"1•11111•11 Ill 1!1 'I II :1111111111~ llJ 111w1111m1~m Ill 1~111111~1 111 I I "I I I' I L q111 l'I 11 II.J U ll+(IP II HI II 11111111 It~ I ,,,. Llt::: 1.u::s ;;:::~;~::~::o:;;:i:::~::t~,;::r:~:;~:$::;;;;;;,·:i:;:::~: @167 @255 @670 u.2:: • t..u::: ti11•111 1·1111 \1,1~/1,lk111/.\l1"11 1/~Y1.\~1l11\1;\111'•~1\i~f1,1i~Mlw~­·:;~~:;':;::~::~::·:~~s:::::::::'.~t;::~ © ·670 Fig. 7. Reading the C and D Scales. the number sequence 1543 is to be set on the D scale. In Figure 8 we see that this setting will fall between points D and E (between 1500 and 1600). Further, the setting will fall between F ( 1540) and G (1550). Finally, the actual setting 1543 will fall, roughly, 3/10 of the distance between F and G away from F, or 7/10 of the distance be­tween F and G away from G. Thus point H represents a setting of 1543. It should be noted that, in the range 1 to 2 on the C and D scales, a sequence of three significant figures can be set in the rule. Thus, if the sequence of numbers is greater than three, interpolation is necessary. Furthermore, in the range 2 to 10, only a sequence of two significant digits can be set. Thus, in this range, if the number of significant digits is greater than two, you must again resort to inter­polation. However, the fundamental rule to remember here is: "The posi~ tion of a number on the C and D scales is independent of the decimal point." 18 .---D-1500 G-1550 F-1540 Fig. 8. Interpolation on a Slide Rule Scale. Exercise 1 1. Make a sketch of the C or D scale indicating the general method of dividing the scale. 2. On the sketch of problem 1, indicate the main numbers of the scale ( 1, 2, 3, etc.) . 3. In Figure 9 give the slide rule setting for the positions A through L, inclusive. Give as many significant figures as possible with one digit of interpolation and, of course, neglect decimal points. VIII. Multiplication As the indexes of the scales play an important part in the multipli­cation operation, it is worthwhile to recall at this point the location of the indexes. The left index of the C or D scale is the line at 1 on the extreme left end of the scale. The right index of the C or D scale is the line at 1 (or 10) on the extreme right end of the scale. The A and B scales actually have three indexes each, namely, a left index, a cen­ 19 ter index and a right index. The first range of scale A and the first range of scale B is that complete set of numbers between the left in­dex and the center index of each scale. The second range of scale A and the second range of scale B is that complete set of numbers be­tween the center index and the right index of each scale. Although the extreme left and right ends of CF and DF scales frequently are referred to as the adjustment indexes, the true index is the line at 1 near the center of the CF and DF scales. See Figure 10 for the loca­tion of these indexes. Multiplication is ordinarily accomplished on the C and D scales; however, the A and B scales, and the CF and DF scales are also use­ful for this purpose. The process of multiplication can be accomplished in three steps: Fig. 9. Problem No. 3. (Porl A-EJ 20 -.­ • I Cl c _,_ ® ----+----.,l 0 Fig. 9. Problem No. 3 (Part F-LJ 1) the first number is set on the D scale by adjusting the slide so that one index of the C scale coincides with the number, 2) the cursor is then moved until the hairline coincides with the second number on the C scale, 3) the answer is read on the D scale under the hairline. These three steps are illustrated in Figure 11 where the operation of multiplying 22 X 3 is shown. The first step is to move the slide until the left index of the C scale coincides with 22 on the D scale; the sec­ond step is to move the cursor until the hairline coincides with 3 on the C scale; the third step is to read the answer ( 66) under the hair­line on the D scale. It should be noted that, in some cases, the left index of the C scale is moved to coincide with the first number on the D scale while, in other cases, the right index is moved to coincide with the first num­ber on the D scale; thus, in multiplying 52 X 3 (Fig. 12), the right 21 ADJUSTMENT INDEXES, CF AND OF SCALE INDEX, CF SCALE INDEX, OF SCALE LEFT INDEX, C SCALE RIGHT INDEX, D SCALE RIGHT INDEX, C SCALE Fig. IO. Location of Indexes• index of the C scale is moved to coincide with 52 on the D scale, the indicator is moved to coincide with 3 on the C scale and the answer (156) is read on the D scale below the hairline. In other words, in multiplication it makes no difference whether the left or right index of the C scale is employed as long as the answer falls within the range of the D scale. This means that the operator should use a little "men­tal forethought" to decide which index to use. Of course, with prac­tice, the selection of the correct index becomes almost automatic, Note that in the above operations the decimal point location was neglected. In Figure 11 , we multiplied 22 X 3; however, if we had 22 DI OF multiplied 0.22 X 0.3, 0.22 X 3, 2.2 X .3, or any other arrangement with the same combination of numbers, namely 22 and 3, the answer we read from the D scale would still have been 660. In short, we have to determine the placement of the decimal point in every operation. How to find the correct position of the decimal point will be discussed later. Had we so desired, we could have used the A and B scales to mul­tiply. In this case, the operation would be exactly the same as before with the B scale (on the slide) corresponding to the C scale and the A scale (on the body) corresponding to the D scale. At one glance, it will be seen that the A and B scales are similar in nature to the C and D scales except that the range of the A and B scales is twice that of the C and D scales. This double range for the A and B scales has certain advantages and, also, certain disadvantages. The "mental forethought" neces­sary to determine which index to use when employing the C and D scales is not necessary here. Suppose, for example, we wanted to multiply 135 X 8; Figure 13 shows the left index of the C scale set on 135 on the D scale. We can see immediately that when we move the hairline to coincide with 8 on the C scale, the answer is out of range of the D scale. Thus, we have chosen the wrong index and in order to solve the problem by using the C and D scales we must move the slide until the right index of the C scale coincides with 135 on the D scale. Then we move the hairline to coincide with 8 on the C scale and we read the answer ( 1080) on the D scale. Thus, in order to multiply these two factors on the C and D scales, only the right index of the C scale can be used. 23 This problem can be avoided by using the A and B scales, although multiplication is not the primary use of the A and B scales. The only rule to remember in order to avoid running out of range of scale A when multiplying by use of the A and B scales, is to set the left index of scale B so that it coincides with the first number in the first range of scale A, or conversely, to set the right index of scale B so that it coin­cides with the first number in the second range of scale A. Thus Fig­ure 14 shows the multiplication of 135 X 8 using the left index of scale B while Figure 15 shows the same operation using the right in­dex of scale B. Of course, it can be seen that the accuracy of multipli­cation when using the A and B scales is less than that when using the C and D scales since the length of one range of the C and D scales is twice the length of one range of the A and B scales. However, the center index of scale B can be set in either range of scale A. 24 1''l''''l111 1l1111 1'1fi1'' '3''''1''''1''''1''''1''''1''''1 I I I I ' I I I I~ I I 'I'''' I~ I 'I'''' I~''' I""I~"' I""I~"' I'"'I~ Fig. I 5. Multiplication on the A and B Scales. Also, the CF and DF scales (sometimes called the folded scales since the range is folded at r. ) can be used to multiply, the CF scale corresponding to the C scale and the DF scale corresponding to the D scale. Figure 16 indicates the use of the CF and DF scales to multi­ply 135 X 8. This operation is, again, accomplished in three steps. 1080••----­ Fig. I 6. Multiplication on the CF and DF Seo/es. The index of the CF scale is placed to coincide with 135 on the DF scale, the cursor is then moved until the hairline coincides with 8 on the CF scale, and the answer ( 1080) is read beneath the hairline on the DF scale. Also, for rapidity of calculations, particularly when the C and D scales are used and the wrong index of the C scale has been chosen, the use of the CF and DF scales in conjunction with the C and D scales is recommended. Suppose, in the earlier problem, we had at­tempted to multiply 135 X 8 and had used the left index of the C scale to coincide with 135 on the D scale. At one glance, of course, it can be seen that when the cursor is moved until the hairline coin­cides with 8 on the C scale the answer is out of range of the D scale. Figure 17 shows that, when the wrong index of the C scale has been LL02 f, 11f.111I1111I11111111:'r.111l1111l1111l111~111!11ul1111h1;~111fu,.l1111lu 1 111l111~r.11d111d1111l11u ,;~ri111\11ul1+11I uhl LL03 ~'""~'~"'"'"'"'''"f,,,1,,,.1,,,.1,,,,1,,X,,,,,,,,,,,,,,,,,,,,,,~,,1,,,,,,,,,,,,,,,,,.'f,, , ...1......... 1 ........f........1r.... ~....... :r.•.. '"' 0 F t1l111tlm1fnul1ml11olnttlnnht1~ ~I l~/11,l~~A(J,IJl'lt\Jihfil.Ji\Alib~\l{\ 111/ \1,1/t'j1 .;'ll\11\11i1j1!1iW1~~1ft~, r, .l/o ·~ W1~·l'~l·~,Mll 1)i1 CIF •""'""$"'1""'""'"'''""1''''•'••·•···· •''61""'"'i)'"'l'".i ...,... • "'i'""''"'''" ,,,, Cl LLl !"'J111~1·mrtttuttt LL2 fig. 17. Multiplication by Use of Cf and Of Scales Without Moving Slide When Wrong Index of C Scale is Selected. chosen, the problem can still be solved without moving the slide. This can be done, as shown, by moving the cursor until the hairline coin­cides with 8 on the CF scale, and the answer ( 1080) can be read under the hairline on the DF scale. However, the primary function of the CF and DF scales is the op­eration of multiplication by r.. Actually, this can be done with no par­ticular slide setting. The procedure is simply to set the cursor until the hairline coincides with the multiplier of r. on the D scale, and the product (answer) is read under the hairline on the DF scale. The same relationship existing between the DF and D scales also exists between the CF and C scales in that any number may be multipl~ed by r. when the hairline is set on the number on the C scale and the answer is read directly above on the CF scale. Figure 18 shows the 9.42 9.42 17.30 17.30 3 x 1r 3 x 1r 5.5 x 1r 5 .5 x 1r multiplication of 7r X 3 and 7r X 5.5 using the D and DF scales, and also the same operation by employing the C and CF scales. If the product of several numbers is desired, multiply the first two numbers together, then multiply this product by the third number, and repeat this process until all numbers have been multiplied to­gether. In other words, when multiplying three or more numbers to­gether such as 12 X 25 X 8 (Fig. 19), we first multiply 12 X 25 and ll02 I,',,91': 111I111111111l 111~11!11111l111d11;~11d1111luuluX11111111l1111l111~111l1111l1111ln·~11111111I1111I1111I111·?r.,,; 11ult111l111 LL03 ~1li1·1~11hmlnuluull11~ul11ul1u1l1111l11~111l1111l1111lu11l11:f.111l1111l1111l1111l11;~1r1l1111l1111!1111l111~J....a.1l1111hml11ul1ml1111l11;1 1111111~1111 °F T.,,,,,,,,,,,,,,,,1,,~,;'1"~'1W111~,1,1,1+•1,1,t~!~1,11,111 1,·~11 1'"i'I11,,11111 ..11I11,111wl1 \1\'1111\,11!11•11l\\'11\1i1i11~11~w11\l'ITl'lf11 il\'l11i1.1~1,1, CI F 1'"'1" " '"'1""11"'1""1'"'1"" 11'''1''' ' 1''' '1 '''' ""1"fl''"1"ti1'"'1"Jl'"'1"6l""1'"Sl"u1"'41""1'"jl" '1"'21'" ''' Cl th111.1ol.1o1o111""''''''''1o1'""'''''''"'··,,,,,,,J,,,,,,,,,1,,.1.1.1.9.1.1,1~1.1. 1.1.1.1, ,,.1.1.1. 1.1,,, 1,,, "'""''""h"''" D l'''',,'''1;'" ,~. ,~,',\\/l/,\~ri1,11111~1ff1,1q~M\l:,,~1~11ri~ll/hlll~hllllh{i~11~1'l;1,J'll'~\'li1'I; .• 11 11.11111.11111,111! LL 3 ~m1111111mrniu111 1 1 11111111,111,11,.,,1,1,1•1,•111,14'l''111uuru11111.mriH1•"~ •1111111m111111111;is 1i1111 11•1111111•1•riJ 41 LlZ 111111.~111111111 111111p 1111111111111111111;T~"1in•111111m•rm111111m1~~11111111111 111111111•~.'!~''l 1 111111111111111:!~1111111111111 _ , 1 LL02 11lm1l1111h11~111l 1111l1ml11-~t 11I1 11I1111l1111l111·;r.111l1111l1111l1111l11 LL03 11111111~11111111111111111111111 11t1hw1hm1tmlunl1111luulu;~111l11iX111l111· ? OF CF 1!~!1!•I1\•/1l1\•J11M111\.ii11\'11 \•1111~1'11~/ili~lillW~ilil~'"""'''',, '1''' 'I , ,, 'I' ~I.' •• '.''' il\ ~1i11111111'1lfthl/t1n11111111111111111'Ulf3 LL3 l""llllltlljlll.IH)llliH~ Jlllllll "'i~l'.1'111' 'l'l'l'l'jlll!l/l~lf LL2 '*11111111111111u1111~;.1l1~1111111 1111111111111 11111111111t'f' 1. l . •11 300 x 8 Fig. 19. Mu/lip/ieolion of Three Numbers Using lhe C and D Seo/es. when we have this answer (300) we move the slide but not the cursor until the correct index (if the C and D scales are used exclusively) of the C scale coincides with the hairline on the cursor. Then, we move the cursor until the hairline coincides with the third number (8) on the C scale. Finally, we read the answer (2400) beneath the hairline on the D scale. Exercise 2 a. By use of the C and D scales only, perform the following multiplica­tions : I. 16 x 5 6. 0.16 x 0.05 2. 17 x 41 7. 1,080 x 0.003 3. 9.5 x 2 8. 63.5 x 18.12 4. 3.75 x 5 9. 2X6X9 5. 64 x 400 10. 0.2 x 0.7 x 3.5 b. Perform the same calculations as in part a, but use the A and B scales only. c. Perform the same calculations as in part a, but by using the CF and DF scales in conjunction with the C and D scales. IX. Division Division, the reverse of multiplication, is treated as such on the slide rule. As in multiplication, the C, D, CF and DF scales are em­ployed in division. For less accurate work, the A and B scales may also be used. Division is actually more simple than multiplication since it is not necessary to determine which is the correct index of the C scale to be used in order to obtain a reading that falls in the range of the D scale. Division, as is multiplication, is accomplished in three steps : 1) when dividing one number by another, set the hairline of the cursor to coincide with the first number, or dividend, on the D scale, 2) move the slide to such a position that the second number, or divisor, on the C scale is beneath the hairline, 3) read the answer on the D scale at whichever index of the C scale falls within the range of the D scale. Thus, in Figure 20, the operation of dividing 75 by 5 is illustrated. The cursor is set so that the hairline coincides with 75 on the D scale; the slide is then moved so that the reading 5 on the C scale is below the hairline; then, since the left index of the C scale falls within the range of the D scale, the answer ( 15) is read on the D scale directly 28 LL02 . 1111!1111l11~111l1111l1111l11;~,t1l1111l11ul111~•.. "f.111l1111l1111lu11l111~111l1111l111tl1111!111~1tl11t11l11' 1111l1 ..~11l1ml1111lu11l11~ 0 LL03 1l1~11l11ul1111l1111!11;fi11ih111l1111l1111l111~111l1 11 1lu~?1~11l1111lt11??1~11i111?'?~ I I 1 11~1!11t111?~~~d 11~! 1'1'1 11 OF ~f:1'~olo\/ol~loiol/,,,\/,,,\l,1,\1,1M,lo\/i',1 ,':' ~1; J~'I 1lrv1\•1v/~1~I\'1\~yl~1~\11,11~1 11.1111~/,\1,11~111.\111~/,\11.11~111.111~ CIF ..'"1"")""'"'11 " 11"''1 '"'1"''1 1• 111•191•1•1 1111•1•1•1•1•9111•111•111•1'1'1'}''' 111'1 '1 'I I' I 'I '6'11I' I'.' I' I' I' ' CI t.....,1o1.....1o ..?.1or.1o1.l.11111.,.~,,,,,,,,1. '""'""''"'''"''''''''''''''''''',, ,,, '' ""' .......~1.......~1....1. l''''~oof.oootMo\lo\l~hJ~1~!1Mo1/~1,\l.'.J~/,J~l11'.' !·11·i·11•\01j'l'fl'\'1\'l'fl11111'o\111 1•,i,,,i,•,i1,l,'o 0 111 ~I~ l,o,\,\i/.\,1l,1\1,1,1t1 0 6 · 8 I 9 ·1•l~6~1l11111111111111i-n~llJfll I~ .,..,,.,.1.,.,,,.111•1•i1r1•111•1q~••1••••1•• 1~1· · I I 1111111~~111!6 LL3 4 S · I 7 LL2 1111~!~'1' ..~.....,........1"~~··•111111•• •1•1i!'i'''111•r1•1•1•t!~''''''';!~''•1• 'f"''""I 111111 11111•111r11111111rn~ • 5 15~~~~~~~~~~~.. Fig. 20. Division by Use ol the C and D Sea/es. below the left index of the C scale. The division of 378 by 6 is shovvn, also, in Figure 21. Figure 22 illustrates the use of the CF and DF scales in the process of division. In this case, 91 is divided by 13 to give 7. The indicator hairline is set to coincide with 91 on the DF scale, the slide is moved until 13 on the CF scale coincides with the hairline, and the answer 7 is read on the DF scale directly above the index of the CF scale. If it is desired merely to divide a number by 71', the DF and D scales or the CF and C scales are useful since no setting of the slide need be made. Re­gardless of the slide setting, wherever the hairline crosses the D scale is the answer that will be obtained by dividing the number indicated on the DF scale under the hairline by 71'. Thus, 4 divided by r. 1s 1.273, 12 divided by 71' is 3.82, and 8 divided by r. is 2.54. This is shown in Figure 23. •1111~r;111111 I'''"''1111'"';f.,1111 '''II"' ''"""~711.i11111""'"''""~"''111111111111..1i11)-;11.1""'""'"1tl· LL02 1Jiml11111i11f.111l11tr.t1d1II~:111 JI I ;,;hfdd1l.l111ItI:'~lll1t11h111l1111f11~?1~1d1111 I 1 I I ??1~ 11 I 1 I l LL03 9 1 1 2 3 • 15 15 17 18 ,g 2 ~11!1~~11'/h~fJ~ll\~h\/~l,\,\1,w,l11,11~,1i11 /~o11.1J~o,o,1l 1°1•1i1wliw1'.'l1 \'.'lil'i'll'i'l/il1/I 1°11\\'ll l'/llW~'.'11\',X'' ''' 1' OF 1 CF ••••. ,.}·1·1·•· •. 1.•. 1·····s· •• , . , .•. 1.•.. , . , • 5.,.,., ... , .•. I .•.•• 1r............,.........,.........l""'"J'.i CIF 1+,,,1,,,, It,,, uu1u?l11nl11~f,.,,1,.~J,. ""~I, ,.,~l,..,1,,.~l,,,,,,.,~I,, ''I•• .~I.••• I+•• ~I,• +• t • • • • Cl 1111 1111 1c jolo11/ollolo'.lo'1/ol/1\il11olol(olo'oiod1\/olo\l,\/1l.lo llol/olo o'/'11111'l'11111111'o1\111' 1 1~l'o1 1\'1'11 i'\'/~1l'/\ll1'1'/1/\.f,,,' 0 3 4 5 6 '{ LL3 1,' 1'1 '1'1~1•1'1'1'1'1'111'~1111111~ ';l~1111111111111111u11i'6~1111 11'l'l'l'li6~1111111l""l"'*l'To~111 11 1~!~1 1 1 LL2 lllljllUJlllll~'.'~~11'111~ 11 I' I'I' Ir 111111i!$l'l'l'l'f'l'l'lll;i~'I •p1•1111ftl~l.~'l'l'JlllllllP~\~ 111 1 378 ~ 6----63 Fig . 21 . Division by Use ol C and D Sea/es. 29 LL02 1h1;~111f1111l1111f11;r.111l1111l1111lit1~1lll11ut1111l11·1~ 11111411I11111111I1111? llllltlllltll LL03 o111111111111111u~~11,111111i •• ,1i1111.,;~."'''''''''''''''''''}e1111111l11..1....l••1111111l111111111l11111•~""...i~ . 5 6 7 ~ ~ OF 11111 11 CF ,:•l•'11/,\l11i1111.\11\1J11/.1l1w111.\11\j111/.\l,~111 11\/\\h11~1\'/1~111l~M\\'111~Ml 11'11~11, j1115p1111114111111111 jlllII I I ll2111111111 I~ I I 1111 I I rlllllllll'llllllll~lilllllljll"fllll' II 'l'!'l'l'flf1 CIF -4 Cl I 1 I 1I1I1I,I1 I 1 I 1 I 1 lt111l1111h111l11t1l1u1l11u 111lu11lt111l1111 1111l1111l1111l1111l1111l111 I 1111It111I11 c 1 1 11 111 D 1,•,11,1~111.11,\U·'l·111~,1l1.11~1.11.111~1.111,~11 f·,·1111·11·,·1111·111·11111·111·1111'111·1111·~,1., .1,1~1,1,1,1,,,1, LL3 1•1·1~·1•111111A111111111111'1'l'lj,'•l l lllll~llllllll 11111111.111111~1111111111111111111111~~ I 111 •1~1~111•1•1•1• 111111111111111111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111111 LL2 1.2 ~ 1.3 1.3~ 7 ,. 91 -:-JJ:.....__ _. Fig. 22. Division by Use of CF and DF Scales. 8 ..;. 1T = 2.54 4 1T = 1.273 c 0 /,l/11\l/,l/lol //,l/1'.//,\llol/1.i//1'110 /,',//,\/ 111 ,f'/'11/l'\11'/'i'('l\'1'/ll'/'/\'1'\'1'/1/'/',11'1111\/.'1"1• 3 5 111 11111 1 LL3 111 11 11 1 1 l!J,,;1'111 1 j11'1'11~1 ~1;;11 ~~ '1""1'1111~16~111 1 1 1'1'1'1'1~6~1'1""1'"'· LL2 •11'f'M1l~'.lM;ll I I : ~1'1'1 1•1 11 1 1 111~!~1 'llllj1l'l'l'li!~'l'l'l'l111111•1~!j'l'l'l'j'l'l111~ 1 12 ..;. 1T = 3.82 Fig. 23. Division by " Using the OF and D Scales. (Same Relation Exists Between CF and C Sea/es.) Also, of course, the A and B scales may be used to perfonn division although they are not intended to be used for this purpose. The pro­cedure here again can be accomplished in three steps. First, the cursor is moved until the hairline coincides with the first number, or div­idend, on the A scale; second, the slide is moved until the second number, or divisor, is beneath the hairline on the B scale; third, the answer is read on the A scale at the point where any index of the B scale coincides with the A scale. Also, when dividing by use of the A and B scales, any combination of ranges may be used. In other words, the first number, or dividend, may be set on either the first or second range of scale A, and the slide may be moved until the second num­ber, or divisor, is on either range of scale B beneath the hairline. Again, here, the answer is read on the A scale at the point where any index of the B scale coincides with the A scale. Exercise 3 a. By use of the C and D scales, perform the following divisions: 25 125 1. -5. 5 17 76 18 2.-6. -­ 5.85 6.67 102 34 3.--7. 0.010,9 53.8 99.6 1,785 4.--8.-­ 22.3 47.4 b. Perform the above divisions by use of the CF and DF scales. c. Perform the above divisions by use of the A and B scales. X. Placement of the Decimal Point In many simple problems the decimal point can be located in the answer by the method of inspection. However, in more complex prob­lems involving several divisions and several multiplications, it is very difficult, and, in many cases, impossible to locate the decimal point in the answer by inspection. Therefore, a rapid, sure, and simple method must be used to determine the location of the decimal point in more complex problems. 31 A survey of the methods of decimal placement employed by the University Interscholastic League Slide Rule Contestants conducted over a period of several years revealed that the majority of contestants used the "significant digit" method of location. Also, in the opinion of the authors, this method is the most reliable for all types of prob­lems. Therefore, for the above reasons, the "significant digit" method will be followed in this simplified manual. The "significant digits" in a number greater than 1 is a positive number and is numerically equal to the number of digits to the left of the decimal point; on the other hand, the "significant digits" in a number less than 1 is a negative number and is numerically equal to the number of zeros immediately following the decimal point. The following table is given to help clarify the meaning of "significant digits." Table A SIGNIFICANT RANGE OF DIGITS OF EXAMPLE OF NUMBERN NUMBERN NUMBERN 10,000 ~ N < 100,000 +s 83,145.32 1,000 ~ N < 10,000 +4 6,220.19 100 ~ N < 1,000 +3 457.23 10 ~ N < 1000 +2 33.84 1.00 ~ N < 10 +1 2.75 0.10 :;; N < 1.00 0 0.68 0.01 :;; N < 0.10 -1 0.035 0.001 :;; N <0.010 -2 0.004,9 0.0001 :;; N < 0.0010 -3 0.000,52 0.00001 ~ N < 0.00010 -4 0.000,019 The "significant digit" method is employed in the following man­ner: For Multiplication: The significant digits in the answer is nu­merically equal to the sum of the significant digits in the numbers to be multiplied minus one for each time the slide is extended to the right of the body during multiplication operation. Any movements of the slide to the left of the body during the process of multiplication are ignored. Thus, for the following examples: a. 58X300=17,400 Problem (+2) + (+3) = (+5) Decimal Placement b. 45 x 34 = 1,530 Problem (+2)+(+2) = (+4) Decimal Placement c. 920 x 0.05 = 46 Problem (+3)+(-1) = (+2) Decimal Placement In each of the above problems, the slide extends to the left; thus, the significant digits in the answer is numerically equal to the sum of the significant digits in the numbers to be multiplied. d. 35 x 25 = 875 Problem (+2)+(+2)-1 = (+3) Decimal Placement e. 420 x 1,600 = 672,000 Problem (+3)+(+4)-1 = (+6) Decimal Placement f. 80 x 0.12 = 9.6 Problem (+2)+(0)-1 = (+1) Decimal Placement In each of the above problems, the slide extends to the right; thus, the significant digits in the answer is numerically equal to the sum of the significant digits in the numbers to be multiplied minus one. For Division: The significant digits in the answer is numerically equal to the significant digits in the dividend minus the significant digits in the divisor plus one for each time the slide is extended to the right of the body during the division operation. Any movements of the slide to the left of the body during the process of division are ignored. Thus, for the following examples: 250 a. --=50 Problem 5 <+ 3)-(+ 1) = (+ 2) Decimal Placement 0.003,5 b. --= 0.000,05 Problem 70 (-2)-(+ 2)=(-4) Decimal Placement 3.60 c. --=0.40 Problem 9(+ 1)-(+ 1)=(0) Decimal Placement In each of the above problems, the slide extends to the left; thus, the significant digits in the answer is numerically equal to the sig­nificant digits in the dividend minus the significant digits in the divisor. 465 d. --= 155 Problem 3 C+ 3)-(+ 1)+ 1=C+3) Decimal Placement 33 824 e. --= 20,600 Problem 0.04 (+ 3)-(-1) + l = (+5) Decimal Placement 606 f. --= 2 Problem 303 (+ 3)-(+ 3) + l = (+ 1) Decimal Placement In each of the above problems, the slide extends to the right; thus, the significant digits in the answer is one more than the significant digits in the dividend minus the significant digits in the divisor. Exercise 4 Determine the correct answers to the following problems; include the deci­ma! point in your answer. 1. 17.34 x 0.003,57 0.354 7. 2. 192 x 46.8 0.098 0.004,32 3. 37 x 12 8. 0.333 4. 0.004,32 x 0.333 123.4 9. 5. 109.5 x 0.000,018,4 567,890 37 125 6. 10. 12 52 XI. Complex Multiplication and Division Multplication of Three or More Numbers. This operation is merely an expansion of the multiplication of two numbers which was dis­cussed in section VIII. The entire process can be completed without recording any intermediate results. Thus, let us examine the problem of multiplying 52. 7 X 350 X 0.173. The operation is broken down in this manner. First, multiply 52. 7 X 350, and then multiply this product by 0.173. As before, to multiply 52. 7 X 350 the slide is moved until the correct index (right, in this case) of the C scale coincides with 527 on the D scale; the cursor is then moved until the hairline coineides with 350 on the C scale, and the answer ( 18,450) is read under the hairline on the D scale. But why record this intermediate result? The ultimate answer desired is this product ( 18,450) multi­plied by 0.173. So, without ever recording this value, the cursor is left in position and the slide is moved until the correct ( le~t, in this case) 34 index of the C scale falls under the hairline; the cursor is then moved until the hairline coincides with 173 on the C scale and the answer (3190) is read under the hairline on the D scale. Where is the decimal point located? In order to locate the decimal point in the product of a series of numbers the method for simple multiplication is extended. The number of significant digits in the answer is equal to the sum of the significant digits in the numbers to be multiplied minus one for each time the slide extends to the right of the body during the op­eration. Thus, the significant digits in the numbers to be multiplied are 52.7 ( +2), 350 ( +3), and 0.173 (0). The sum of the significant digits in the three multipliers is +5. During the operation the slide extended to the right of the body one time (when multiplying the product of [52. 7 X 350] and [0.173]). Therefore, 1 is subtracted from 5 and the significant digits in the answer is 4. Consequently, the an­swer is 3,190. If more than three numbers are to be multiplied together, the process would be simply an expansion of the above method, with the multiplications performed in pairs on a cumulative basis. Division of One Number by a Product of Two or More Numbers. This operation is simply a series of divisions. The entire process, again, can be completed without tabulating any intermediate value. Thus, take the case of dividing ( 725) by ( 17 X 55). The solution is to divide 725 by 17 and then, in turn, divide this quotient by 55. In order to divide 725 by 17 (a simple division process), the cursor is moved until the hairline coincides with 725 on the D scale; the slide is then moved until 17 falls under the hairline on the C scale and the answer ( 42.6) is read on the D scale at the point where either index of the C scale falls in the range of the D scale. But, again, why record this inter­mediate reading? The answer desired is this quotient ( 42.6) divided by 55. Thus, without recording this intermediate value the cursor is left in position and the slide is moved until the second divisor ( 55) falls under the hairline on the C scale. The answer (0. 775) is read on the D scale at whichever index of the C scale falls in range of the D scale. Where is the decimal point located? In order to determine its location in this case, the rule of decimal point location for simple division is expanded. The significant digits in the answer is equal to the significant digits of the dividend minus the sum of the significant digits of the divisors plus one for each time the slide extends to the right of the body during the entire operation. Thus, in the above ex­ 35 ample, the significant digits in the dividend ( 725) is +3. The sum of the significant digits in the divisors, 17 (+2) and 55 ( +2), is +4. During the operations, the slide extended to the right of the body one time (when dividing 725 by 17). Therefore, the significant digits in the answer is 3 -4 + 1 =0, and thus the answer is 0.775. If the original problem had three or more divisors, the process above would be continued in more or less cyclical operations until all divisions had been completed and the final answer obtained. . . 82X.037X432 Extended Operations. Example: Fmd the result of-----­.54X8.63 X 127 This problem is broken down into simple operations as shown be­low. The method of alternating multiplication and division is recom­mended, where possible, rather than a series of multiplications fol­lowed by a series of divisions. The reason is that a multiplication following a division often does not require a movement of the slide and, thus, this procedure will consume a minimum amount of time. Therefore, the above problem would be performed in the following steps: 1) Divide 82 by 0.54. 2) Multiply the quotient obtained in ( 1) by .037. 3) Divide the product obtained in (2) by 8.63. 4) Multiply the quotient obtained in (3) by 432. 5) Divide the product obtained in ( 4) by 127. 6) Read the final sequence of numbers in the answer on the D scale. Each of the above operations is a simple multiplication or division of two factors. *Thus, 82 I. -= 1,518 54 2. 1,518 x 37 = 561 561 3. --=651 863 4. 651 x 432 = 2,815 2,815 5. ---= 2,215 (the sequence of numbers in the answer) 127 * Note : Only sequence of numbers is determined here; decimal point is ignored. To determine the decimal point location in the answer, the rules for simple multiplication and division are extended. Operations ( 1), ( 3) , and ( 5) are simple divisions. In two of these operations, ( 1) and ( 5), the slide extends to the right of the body. Operations ( 2) and ( 4) are simple multiplications. In one of these operations, ( 2) , the slide ex­tends to the right of the body. The sum of the significant digits of the numbers in the original problem above the division line is + 2 -1 + 3 or + 4. The sum of the significant digits of the numbers below the division line in the original problem is 0 + 1 + 3 or + 4. There­fore, the significant digits in the answer is equal to 4 -4 + 2 (two extensions of the slide to the right in division) -1 (one extension of the slide to the right in multiplication) . Consequently, the significant digits in the answer is + 1 and the value is 2.215. Actually, when performing an operation such as the above example, each time the slide extends to the right during multiplication a -1 should be recorded and when the slide extends to the right during division a + 1 should be recorded. These can then be mentally summed to find the total correction for slide movement to be applied to the significant digits of the original numbers in order to determine the significant digit value for the final answer. There are many complex combinations of numbers similar to the above that can be solved by simply an extension or a slight variation of the above methods. Investigate the problems given in the following exercise to test your ability to solve a few such complex arrangements. Exercise 5 Determine the answers to the following problems by use of the C and D scales. Be sure to indicate the decimal point in your answers. 36X51X17 48.3 x 17.7 x 39.3 J.-----6.------------~ 25 x 16 x 37 0.000,733 x 0.000,044,1 x 96,400,000 96.1 x 48.5 x 0.033 0.012 x 0.033 x 41.4 x 76.3 2.--------7.----------­ 21.7 x 16.4 x 45 36.2 x 0.000,339 x 0.442 0.002,77 x 0.046,2 414 x 2.41 x 0.017,8 3. -------8. ---------­ 0.003,66 0.013 x 0.016 x 9.5 x 48.2 484 x 5.99 x 63.3 40,000 x 58,100 x 0.061,3 4.-------9.---------~ 41.2 x 0.067,7 0.002,4 x 78,000 x 209 0.098,8 x 0.136 x 44.4 200,000 x 48,100 x 0.000,072 5.---------J0.----------­ 100.4 x 0.007,7 x 39.5 48.2 x 0.000,393 x 750,000 37 XII. Squares and Square Roots The determination of squares and square roots on a slide rule varies depending upon the scales appearing on the rule. In general there are two different types of square root or square scales used, namely, the A and B scales, and the R1 and R~ scales or two y-scales. For Rules Having A and B Scales. The A and B scales are identical in calibration, just as the C and D scales are identical and also the CF and DF scales are identical. The A and B scales each consist of two D scales that have been reduced to half length and placed end on end, with the A and B scales each having approximately one-half as many calibration lines as the D scale. If one desires, he may multiply and divide by use of the A and B scales; however, as mentioned earlier, the accuracy of his answer will be less than that obtained by using the C and D scales. This is not the intended use of the A and B scales. In order to determine the square of a number, slide the indicator hairline until is coincides with the number on the D scale and read the square of the number directly under the hairline on the A scale. See Figure 24. If the number to be squared (set on the D scale) 1s 2 2 31 = 961 31 =961 g I\' !i\'j'/i'l\11'!11p'l11\jl\' 0 DI HJlllTll '"'I'" 1'7fil'~'!'"'l,'''l''''I '' 1111'' 'I''''I' I 11 I' I I' I'''' I' ~· 2"'1':;1111111~~1'111111: OF .~ 111111111~" I '"'l~"'ltl"J;"'l'll'l~lll111111;11•11111l~"'l"lll~11111•2··'I I' I'' 1·'.I11 11111 ... ,, Fig. 24. Squares and Square Roots Using the A and D Scales and also the B and C Sea/es. greater than one and the answer (read on A scale) falls in the first range of scale A, there will be an odd positive number of significant digits in the answer. For example, (Fig. 24) when we set 31 on the D scale we read 961 under the hairline in the first range of scale A. Therefore, we know the answer will either be 9.61, 961, 96100, etc. Logical reasoning tells us the answer for 3l2 is 961. Conversely, if the number ( 55) to be squared is greater than one, and the answer ( 3025) falls in the second range of scale A, there will be an even positive number (+4) of significant digits in the answer. If the num­ber ( 0.02) to be squared is less than one, and the answer ( 0.0004) falls in the first range of scale A, there will be an odd negative number (-3) of significant digits in the answer. If the number (0.06 ) to be squared is less than one, and the answer (0.0036) falls in the second range of scale A, there will be either zero significant digits or an even negative number (-2) of significant digits in the answer. Determining the square root of a number is the reverse of the above procedure; however care must be taken to locate the number on the correct half of the A scale. If it is desired to determine the square root of a number greater than one and having an odd number of digits before the decimal, such as 9 or 125, slide the indicator hairline to this number on the left half of the A scale and read the square root directly under the hairline on the D scale. Ifit is desired to determine the square root of a number greater than one and having an even number of digits before the decimal, such as 25 or 4900, slide the indicator hairline to the number on the right half of the A scale and read the square root directly under the hairline on the D scale. If it is desired to obtain the square root of a number less than one and hav­ing an odd number of zeros between the decimal and the first signifi­cant digit, such as 0.04 or 0.0009, slide the indicator hairline to the number on the left half of the A scale and read the square root of the number directly under the hairline on the D scale. To obtain the square root of a number less than one and having either no zeros or an even number of zeros between the decimal and the first significant digit, such as 0.25 or 0.009, slide the indicator hairline to the number on the right half of the A scale and read the square root directly un­der the hairline on the D scale. The decimal point is determined by reversing the procedure for squaring a number. The above procedures may be applied to the B and C scales exactly as described above. The number is set on the C scale and the square is read on the B scale. This permits square and square root determina­tion on the slide independent of its position relative to the body. For Rules Having R1 and R 2 or Two yl--Scales. Note that the Ri and R2 scales (or the two y~scales) are similar to one-half of the D scale; the R1 and R2 scales (or two y-scales) placed 39 with their ends together would be similar to a complete D scale but would be twice as long as a D scale. To determine the square of a number, move the indicator hairline to the number on whichever of the R1 or R2 scales (or the top or bottom y-scale) on which the number appears and read the square directly under the hairline on the D scale. Note that the slide is not used and could have been removed from the rule if the operator so desired. The decimal point is determined either by reasoning or by approximation. Determining the square root of a number is the reverse of the above procedure; however care must be taken to locate the square root on the correct scale (either the R 1 or R 2 scale or the first or second y--scale) . If it is desired to determine the square root of a num­ 1 2 eJt LLO l.Of,1,,,1,,,,l,,,,1,,,,l,,,,1,,.,l,,,,1 .. ,,l.. ,.1,~,?P,~,~1,,., .. ,.1 ....1,. .. • ... l .... 1 ,,,.1,.,,1,.,,1,~.'f.~ ~ LLIO ·91~'' I I1'' 'l. I II I1111I.I"I11 .. I... d1111l111d1~~1~~.1.......1....1.... 1....1....1.... 1.,.. 1:~ ~.''I 1 I I' I1'' .1, I' .i .. " l... .t .. 11I111 .1., .. 1,;~~~. K l1111l11ul1ml1111lnuluul111~11..l..iJ1l1l1lol1l1l1l1l1l1f1Md1l1l1t.l1l1l1 ,M.iJ.iJ,,,,1,,,,J,.,,1,,.,l,,,,1,,, ,,,,1,..,l....a.!f.,,,1,.,,1.,11l1111l;,11!1111l11wl~1l1l1l1l 1l1l1l1l1' ~: ~1J1111J1111J1111!11t1~111J1111J1111J111i1111111111111J 1J1 It 1J1J11111!111111111l11111 1111111111111!111111111i1lt11111~111111Jii11111111~111111111i1 cIF !i11111"~'"1""1''"1''''1''''1' 11'1'''' I'' 11 I''I 11' I,, 1 1"1"~1""1"~1'"'1"~1""1"~1 "'111~1""111~1''''1''';1··"1'";1···' I'' 'i. I'''' I' c I l1.1i1.1,1.1.1,1.1J.1,1.1.1.l.1.1,1.1~.1.1.1,1,fi1,1,1i1.l1.1.1.1.1.1.1 1.1.~.1.1.1.1.\,1.1.1.1 •• 1. 1.1. I I J, 1. 1. t .1...1....1....1....1.... 1....1.... 1 .. ~ !1111J1111ii111J1111i!111J1111i!111J1111~111J1111j~1111111 ~~111i~1111111j~11111111[11111111111 1111J1111J1111J1111i1111J1111J1111J1111J1111!111 Rt j' I' I 1 I'I'I1 I' I' I' I' I~ I' I' J '1 1 I'I'l1I1I'l~l1I'1 1 I'I11 I 1;1•1 1l'l'l'l'l'l'l'J:I I l'l~l'j;l'l'l'l'l'1'1'1'1'1~'1'1'1'1'1'1'Pl'li'l'l'l'j Rl ' 111j~"l""l~"P"'l~"l""l~"l'111l~"l""l~"l""l~"l"11l~"l11" "'I" •; 11l''"l~"l""l'~"l""" """J' l""l'~'l'"T;'l""l;'I'"~'' ''I'''' I''' '1 1'' 11''''1'' L •"'l""l""l""l""l""l'"'l'"'l""l""j""l""l"'l""l""l'"'l""l'"'l""I""~ '"' "l'"'l""l""l""l'~'I'~':~':"" "'1'"'1""1""1""1""1"'4'"1""1""1""1""1""1'"'1""1'" 16 2 .. .. LLO ...1....1....1... L~.~1....1....1....1.,,.1....1....1..,,,,,, ..~,,1. 1.1. 1.1.1 ,1.\?f.t,1 •• 1.1.i.i.i~?P.7.i.i.i.1.1.i.t.1?P.r.1. LLIO ul11nl1t11lrn1l;~,T.~,l11nlu11lm1lnul..,,l11nl1111l11• .~1~I1 Io I1I 1I1I1Io\,·1~T~l1 I I ol 1! dolol ;~T~.l1lol1ld1l1l;?,Tl1 K •,,.J,,,,1,,,,1,,,,1....l....1,,,,l....,.J..l3~.. 1., 1,,,,1,,,,1,,,,l,,.,1,,..i...i..iJ,1, 1.1.1 l,1, 1.1,1.f.1.1,1.1.l,1,i.1.11.1.1,u,l...J,,,,1. CF CF ;111J1111 i!111 J1111i!111J1111~11J1111i~1 l1111i~11J1111i~1111111!~1111111i[1111i1111J1111J1111J1111J1111i1111 11 1 CIF '11'Pl'l'l'l'l'l'l~l 1Pl'l'l 1l'fll'l'~11'1 11'1' '111 11'11$'1 1111111 11111111 ·~1 1 I' I' I' 11 I' I' I ·411•111 CI 'uul111111111I11•1l1I11I1o11 Io It• Io o ,t.,,1,!l1ml11~l11111o1?.luul11~ 1111l11~l.11d11~L It 1lno~J,1o 1 I1 11~ c 0 R1 tllj1tlljl1tlj1111j111! 1J1J1J1J1!1[1J1J1 l~'l'l'l~tp1•111•1•1•1•11 ·I '' I' I'' I~ 111 !'II' I It I iJtl1\1J1j1JiJ1l1Jtl J 1!j1J111l1JtltJ1J1jil111J1J~J1j1;• • 1;111111" I~"• 1111 I ; "' l~'tll"" 1;•11111111~··11•+ R> • · 1A''"l""l'"'l""l'uT111p111p111p11111111 1111111 1•111l"''l'"T'"l"''l"n 111 1rn11111111"l""l"11111111 1111111111111~1111 L """l""l""l""l"'.i"'l'"'l'"'l""l""l""l""I""~~ "" "l""l""l""l""l""l""l";~"'I' l""l""l""l"''l'"'l""l""l"'r'I"" 4900 625 Fig. 25. Squares and Square Roots by Use ol R and RScales. 1 2 her greater than one and having an odd number of digits before the decimal (such as 2 or 625), set the indicator hairline on the number on the D scale and read the square root under the hairline on the R 1 scale (or first "\;-scale). To determine the square root of a number greater than one and having an even number of digits before the decimal (such as 16 or 4,900), set the indicator hairline on the number on the D scale and read the square root under the hairline on the R2 scale (or second y-scale) . To determine the square root of a number less than one and having an odd number of zeros be­tween the decimal and the first significant digit (such as 0.0+ or 0.0009), set the hairline on the number on the D scale and read the square root under the hairline on the R1 scale (or first y scale) . To find the square root of a number less than one and having either no zeros or an even number of zeros between the decimal and the first significant digit (such as 0.25 or 0.007), set the hairline on the num­ber on the D scale and read the answer under the hairline on the R2 scale (or second y-scale). The decimal point is determined either by reasoning or by approximation. Figure 25 indicates a number of squares and square roots that may be obtained by use of the R 1 and R2 scales. Figure 26 indicates similar operations by use of the two v-scales. Exercise 6 Obtain the squares and square roots indicated in the following problems. Be sure to locate the decimal in the answer. 1. y4 6. (19.6) 2 2. yl6 7. (0.031 )2 3. y144 8. ( 17,241) 2 4. yG.915 9. (5.01)2 5. y0.000,25 JO. (0.002,32)2 XIII. Cubes and Cube Roots Determining the cubes and cube roots of numbers is very similar to determining the squares and square roots of numbers except the K or the -ij-scales are used in place of the A, B, the R1, R2 or the y-scales. These scales are used in conjunction with the D scale. For Rules Having a K Scale. The K scale is similar to three D scales that have been shrunk to one-third their regular size and placed end on end. This is shown in Figure 27. 41 1 t I.I. I .1. I ' 1. 1 'I. I ' •• 1. 1 I .1.1. 1. 1. I' 1.1. f. 1. 1.1.1. 1•• 1. 1.1.~.1.1. 1.1. 1 , . 1.1.1. r.1.1 . 1. 1. I ' t'+""l,.J;.."f'o.\...,..~..#+r--...i.......l-t+<..J.r--l'.+++f...,,.......t+¥-tJ-'I"""'~~~~~~~ I ~~ ] 1 11111111ijltt1!u111111111111111111111t111 111111 11111 11 111!1111111111 111/j 1111ll/111111!111111111l111111111i111111 11111111111111111111~11111111111 11 T~""""l+lfM~~''"'/ol.i,w.~~WlfW~"t'""~~'°l"!l"'i..,.i,+ir....i.i:~~.i,.,t'l"'i-/*T,10;:i-'i....,... - s: ~.TI,~M'T'Mi'J,~rl'rif'!+r'f:i:b~l'l'rt.rn"""1~.tJ:t:,.'"'r'""(""'t'.+f""1""'1""1""'J'"°t':r.-r"'"'t""1"'1....,... 11 Cl 4l,1J.~t.l.1,1,1~.f.1.1,1.1.l.1,1.1:i.f.1!1 , 1.1.l.1.1:1.f.L.1.1 . 1 ,l.1.1.1. ~.f.1.IJi_1. l.1.1 . 1. I~ , I. I. I , I . I. I, L I ,; c l ' f 1.414 2 0.25 0.5 1,1.1.1.l,1.t,t.t f.1 I . I 1.l.1.1 . 1. ,~1. 1.l,l,l,t.l.l ,l ,l~l.t.!,l,l.l,l1l,l,~l,l,l.t,f.l,l,l.l,l~1 l.L.1,l.1,1,1,1,~ • • •I •• · f, ·•• I ,.tu... J 1 I J ,..., . .,,,,..,...,_..,......,.+..r-.,i...#,.-..,~~..,..~l".ltH