Manual / Guide · 1939
Lawrence's own instruction book for its A B CI C D K rules: the 'mental survey' and 'digit and integral digit' methods of placing the decimal point, multiplication, division, squares, cubes and their roots. 16 pp.
‘Lawrence’ | , SLIDE RULE EVERYONE WHO DOES FIGURING NEEDS A SLIDE RULE SS ys jothe i ; 2 Lawrenee’ Sy Ni BY SLIDE RULE i Gr - COMPLETE INSTRUCTIONS ON HOW TO MULTIPLY, DIVIDE. SQUARE, CUBE, EXTRACT - SQUARE AND CUBE ROOTS ON \ THE TIME-SAVING SLIDE RULE SE gee ines SIMPLE RULES for using the SLIDE RULE By John Poland Prof. of Mechanical Engineering and Air Conditioning at the Chicago Technical College FOREWORD Anyone from a seventh grade student up can learn to use a slide rule properly with just a little study ad practice. There are two methods of learning to use the slide rule: 1. The Mental Survey Method ° 2. The Digit and Integral Digit Method The Mental Survey method is quick and practicat. It is used by hundred of thousands of slide rule operators. The Digit and Integral Digit method is more thorough and scienti- fic. It is used in schools and colleges. To the best of our knowledge this is the first time both methods have been made available in one instruction book. If yuu select the ‘Mental Survey" method disregard all material on digits and integral digits. If you use the Digit and Integral Digit method. disregard all material on the “Mental Survey” method. Anyone who will apply himself to the operation of the - slide rule and will make its use a practical habit while at work, study or leisure, will find himself richly rewarded in the sav-~ ing of time and energy—to say nothing of the satisfaction that comes from the mastery of “Man's most useful tool.” Published for Lawrence Engineering Service @ Peru, Indiana — 1939 — COPYRIGHT 1939 BY LAWRENCE ENGINEERING SERVICE—PERU, IND. COPYRIGHT 1940 BY LAWRENCE ENGINEERING SERVICE-—PERU, IND. INTERNATIONAL COPYRIGHT SECURED * ae “THE SLIDE RULE The Slide Rule is a time and labor saving instrument for quickly solving problems involving multiplication, division, proportion, squares, square roots, cube and cube roots or any combination of these processes. The operation of a Slide Rule is simple and anyone with a fair . understanding of numbers can use one. It should be remembered that the Rule itself is accurate within about one-half of one per cent, and the accuracy of the answers obtained are limited only by the spacing of the lines on the scales and the ability of the user to estimate read- ings and setting that fall between the lines. It must also be remembered that only the first three digits of any number can be read-or located on the slide rule except in Units f of the C and D scales, where four digits can be located. A good rule to follow is this: if the fourth digit is less than 5, drop the remaining digits but if the fourth digit is 5 or more, increase the third digit by one and drop the remaining ones. The Slide Rule consists of three parts (See Fig. 1), namely: the rule proper or body containing the A, D and K scales; the slide con- taining scales B, C I and C; and the glass “indicator,” “runner,” oz “cursor” with a hair line in the center. er . : 2 niet ere ‘\ le pave tea tite te aden ale dvated La ENN is {psy i Sigsotararkannadernet a Sink € . fn ae qty : ® Peat hth Scales and Uses Scales C and D are used in multiplication, division and their combinations. Sczles A and D are used for squaring and for finding the square roots of numbers. Scales K and D are used for cubing and finding the cube root of numbers. The C I scale gives the re- - ciprocal of any number on the scale C. It can be observed from Fig | that the scales A and B are identical, scales C and D are also identical and that the C I scale is the C scale in reverse direction. , Before the different operations of multiplication, division, etc., can be pecemplished, the operator must know how to read the differ- ent scales. Scales “Cc” and “TD” From Fig. 1 it will be seen that these two scales are identical © in their divisions and markings. Since most of the operations are based upon scale D we will learn to read this one first, remembering in J aes that the discussion to follow can be applied to scale C. Lookin at scale. Din Fig. 1 we see that it is divided into nine main unequa . divisions, starting at the left with number 1 and proceeding to the “= gight to number 9, thence to a second number 1. =. = ‘These divisions as mentioned before are UNIT divisions and each one is divided into smaller divisions. From Fig. 1 we see that the UNIT 1 (that ‘is from number 1 to number 2 has ten major divisions, each equal to one-tenth (.1) of the unit. These major divisions in turn are divided into ten divisions, each of which is one- tenth (.1) of the major division of one-hundredth (.01) of the UNIT Examples: 17,.15.3, 1162, and 1.92 would be located as shown in Fig. 2. UNIT ONE OF SCALE “D” FIG. 2 Each of these numbers begin with the digit “one” so each will fall nx UNIT 1 of scale D. Number 17 is read at unit 1 plus 7 major divis.ons; number 15.3 is read at unit 1 plus 5 major divisions plus 3 minor divisions of the following major division; number 1.92 is read at unit 1 plus 9 major divisions plus 2 minor divisions of the follawing _ major division. It should: be noted that the decimal point is always disregarded when locating a number on the scale. : ~~’ UNIT 2: Unit 2 is the next main divsion of scale D and from Fig. _» fl we-see it is divided into ten major divisions as is unit 1 but they ar). are not numbered here. Each of the major divisions of this unit are ' _. divided into 5 minor divisions, each of which is one-fifth of two- tenths (.2) of the major division or (.02) of UNIT 2 z ian of UNIT 2: Locate numbers 21.6, .283 and 255. (See 16 255 .283 oi 5: ef UNIT TWO: UNIT THREE FIG. 3 Each of these numbers begins with the digit 2 so it will fall “within unit 2. The number 21.6 is read at unit 2 plus 1 major division . poe 3 minor divisions of the following major division as each line -here equals two-tenths (.2); the number .283 is read at unit 2 plus 8 major divisions plus 1! minor divisions of the following major division; the number 255 is read at unit 2 plus 5 major divisions plus — os , 2% minor divisions of the following major division. Here again- the decimal point is disregarded when locating a number on the scale. 4 UNIT 3: Unit 3 of the D scale is divided as is unit 2 and the divi- sions possess the same values. See Fig. 3 for comparing unit 3 and unit 2, . Eanes for UNIT 3. Locate numbers 3.08, 37.1 and 359. (See. The number of 3.08 is read unit 3 plus 0 major divisions plus 4 minor divisions of the first major division; the number 3.71 is read unit 3 plus 7 major divisions plus 44 of a minor division of the follow- ing major division; the number 359 is read unit 3 plus 5 major divisions. plus 444 minor divisions of the following major division. UNIT 4: Unit 4 is cur next main division and from Fig. 1 we see that this unit is divided into ten major divisions as in the preceding units. Each of these major divisions is in turn divided'into two minor divisions each having a value of one-half or .5 of the major division or .05 of unit 4. When we have a three digit number in this unit the third digit must be estimated unless it is a 5, Examples: Number 435 is read unit 4 plus 3 major divisions plus 1 minor division; number 43716 is read unit-4 plus 3 major divisions plus | minor division which gives 435 and then add an estimated two- fifths of the next minor division giving 437. The remaining digits, 1! and 6, are too small to be considered so they are ignored as explained under the portion headed “THE SLIDE RULE.” See Fig. 4 for these and other readings. : 7 43716 UNIT FOUR FIG, 4 The remaining units on the scale are the same as unit 4 except that the division lines are closer together. Examples for units 5, 6. - 7, 8 and 9 are shown in Fig. 5. ; Woarecteteane pest The number of 5.32 is read unit 5 plus 3 major divisions plus an estimated two-fifths of the next minor division; number .808 is read unit 8 plus 0 major divisions plus 1 minor division which gives 805 and then add three-fifths of the next minor division giving 808; number 9622 is read unit 9 plus 6 major divisions plus two-fifths of the next minor division. (The fourth digit 2 is ommitted on the scale as pre- “viously explained.) a Pe Sn eee Se, ee ere Se OT eee ee aN tnd [5 I ee a Pe ae r Au aa a Bee ee pe Cah t hta deck honedid ou TLL archon Tanah Mek we ‘ MULTIPLYING 14 x 46 FIG. 6 LOCATION OF DECIMAL POINT IN MULTIPLCATION BY THE MENTAL SURVEY METHOD Multiplication As stated before, scales C and D are used in multiplication and are identical. Therefore. the divisions and reading of scale C will be the same as those found on scale D. The first line, number 1, at the left of the C scale is called the left index, while the last line at the right end of the scale is called the right index. When multiplying two numbers together we set either the right or the left index of the C scale directly over one of the numbers on the D scale: move the indicator so the hair line is over the number on the C scale: then read the answer under the hair line on the D scale. , Example 1: 3.4 x 7.2 Set the right index of the C scale over 7.2 on the D scale, move fhe indcator until the hair line is over 3.4 on the C scale Read the 24.5 under the hair line on the D scale. « The most convenient way to locate the decimal point is to make a mental multiplication of the first digits in the factors. Then place the decimal point in the result so that the value is nearest that of the mental multiplication. Thus, 3 and 7, the first digits of our factors will give 21, indicating that the result 245 will become 24.5 instead of ' 2.45 or 245, as 24 is the nearest to our mental multiplication value of e —4— ; Example 2: 14 x 46 = 644 Set the left index of the C scale over 14 on the D scale, move the indicator so the hair line is over 46 on the C scale. Read the answer 644 under the hair line on the D scale. (See Fig. 6.) From mental multiplication we see that 14, which is near 10, and 46, which is near 45, would give 10 x 45 or 450, showing that the answer would be in the hundreds, namely 644 instead of 64.4 or 6440. Example 3: 18.72 x .356 = 6.66 Set the left index of the C scale over 1872 on the D scale, move the indicator until the hair line is over 356 on the C scale. Read the answer 666 under the hair line on the D scale. (See Fig 8.) From observation we see that we are multiplying practically 29 x 3 giving 6. Therefore, our answer becomes 6.66 instead of ‘ 66.6 or .666. Multiplication by the Integral Digit Method. (See page 21.) As stated before scales C and D are used in multiplication and are identical. Therefore, the diyisions and readings of scale C will be the same as those found on scale D. The first line, at number 1 at the left of the C scale, is called the left index, while the last line at the right end of the scale is called the right index. When multiplying two numbers together we set either the right or the left index of the C scale directly over one of the numbers on the D scale; move the indicator so the hair line is over the other num- 7 oa the C scale; then read the answer under the hair line on the scale, ; Example 1: 2 x 3 = 6 Set the slide so that the left index or number 1 of the C scale is directly over UNIT 2 on the D scale. Move the indicator so that the hair line is over UNIT 3 on the C scale, then read the answer 6 ~ directly under the hair line on the D scale. (See Fig. 7.) Example 2: 14 x 46 = 644 Set the slide so the left index of the C scale is over 14 on the D scale. Move the indicator so the hair line is over 46 on the C scale, then read the answer 644 under the hair line on the D scale. (See - Fig. 6.) MADE HU, 8. A. LAWRENCE ENGINEERING SEEVICE, PERU, INDEANA, Lr: CPP APA RARER A RRR 1g satel aRacaatine tt Ce re ee C8 WsutteaPiaiataieRiadabnnttdntaettr tdotRastsnratat sotbatstetat Bots t ety Rabel Che ee er ket a maka A Nala A PP ea i a i aa A Ba SO esretlntontaatabnidcdedod tf TOR PP re ee ee tt de tebe dbete Rd perteet - MULTIPLYING 2 x 3 FIG. 7 oo thesé operation and that the number of integral digits in each answer ‘contained one less integral digit than the sum of the integral ‘digits in both nuribers. From this explanation we can establish a rule for _. + multiplication when the slide extends to the right. RULE: When the slide extends to the right, add the number > of integral digits in the multiplier to those in the multiplicand and then subtract one; the result will be the number of integral digits in the answer. . Example 3: 18.72 x .356 = 6.66 Set the slide so that the left index of the C scale is over 187 on the D scale. Move the indicator so that the hair line is over 356 - om the C scale, then read the answer 666 under the hair line on the . - Dscale. (See Fig. 8.) : LAMONTE CHGNEDING 60 7, NOAA, eu Pactra dissetnibirsatanti ars tga Prete drtseDtteteted tots ts ts Retebetedacendeedet amen LE DT Diesprratrneteta taht Pita ti edd pectet at fetonttte MULTIPLYING 18.72 x .356 FIG. 8 The number 18.72 has plus 2 integral digits and .356 has zero integral digits; adding we get plus 2-integral digits. Since the slide ex- ~- tended to the right we subtract 1 leaving plus one integral digit, thus 666 becomes 6.66. Example 4: 2351 x 41.2 = 96800 The left index of the C scale is set over 235 on the D scale; the hair Hine of the indicator over 412 on the C scale; read the number 968 under the hair line on the D scale. Number 2351 has 4 integral .. digits and number 41.2 has 2 integral digits; adding we obtain 6 integral digits. Since the slide moved to the right we subtract one _. leaving 5 integral digits in the answer. As we are only able to locate. three numbers on the scale (three numbers are all that we can read in any answer), zeros must be added to complete the necessary , number of ineegea digits. We will need two in this example making the answer 96, Problems: 183.2 x 27.42 Ans. 5020 31.52 x 19.7 Ans. 621 1.248 x 6.57 Ans. 8.20 ft will be hoted that. the slide extended to the right in both of | 257 x .313 ‘ Ans. 80.4 .037 x 228 Ans. 8.44 233.42 x .60196 Ans. .457 - It will be noted that the preceding problems in multiplication were such that the slide always extended to the right. We will now use ‘numbers so that the slide will move to the left. Example 1: 7 x 6 = 42 Set the right index of the C scale over 7 on the D scalé, move the indicator so that the hair line is over 6 on the C scale, read the answer ° 42 under the hair line on the D scale. (See Fig 9.) 3 weld ed det f Perrckaneehdttdouiebihebtidded cree: tidasstasditntelanadsslenbiabutebudreforedade n PAT. PEND, Ci Ii Witla Hy SaltrududPtePataat ? 6 a | wicdarfieptiseelas ‘ait anitiniitutant s ditt te ae t t a Tp gp eed Pe a a Ty abinctiay Pevsts .* | rl edadadeLetalatatTnatebsttdsitd re catiabaabuduaduderrdirdnadus MULTIPLYING 7 x 6 “FIG. 9 Example 2: 8.3 x 27 = 224 Set the right index of the C scale over 83 on the D scale, move © the indicator so that the hair line is over 27 on the C scale, read the. answer 224 under the hair line'on the D scale. (See Fig 10.) § RENCE ENGINTIING SEXVIC, PERL INDIANA. PAT. PEND. y ”’ f hutad blaahetniactorky tindacdinbichiebsch inttootithustunl i MULTIPLYING 8.3 x 27 FIG. 10 Example 3: 9.5 x 4.8 = 45.6 . In the above example the slide extended to the left and the’ ’ answer has as many integral digits as there were in the numbers ‘themselves. Hence the rule: WHEN THE SLIDE EXTENDS TO “THE LEFT, ADD THE NUMBER OF INTEGRAL DIGITS IN aon 7 wee THE MULTIPLIER TO THOSE IN THE MULTIPLICAND AND THIS SUM WILL BE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER. | Example 4: 75.3 x 6.27 = 472° Number 75.3 has 2 integral digits and 6.27 has one integral digit, adding, we will have 3 integral digits in the answer; thus 472. Example 5: 4387 x .0372 = 163 . Number 4387 has 4 integral digits and .0372 has minus: integral ae adding 4 to —-1 we have 3 integral digits, thus the answer Example 6: .0057 x .0244 = .000139 Number .0057 has minus 2 (—2) integral digits and .0244 has minus 1 (—1) integral digit. Adding —2 to —1 we have —3 in- tegral digits. Therefore, three zeros must be placed before number 138 making the answer .000138. 75.21 x 485 Ans. 36,500 0672 x 3.48 ‘ : Ans, .234 ° 8.052 x 44.72 Ans. 360 9,122 x .1488 , Ans. 1.357 When multiplying more than two numbers as 6 x 24 x 32, mul- tiply 6 x 24 as has been explained; then take this answer and multiply ee oe obtain the final answer. Multiplying 6 x 24 = 144. 144 x 32 = 4610. In the first operation) the slide extended to the left, therefore the answer will have 3 integral digits since there is 1 integral digit in - aumber 6 and there are 2 integral digits in number 24. In the next operation, the slide extends to the right, so we must use the sum of e integral digits less one. Number 144 (answer of first operation) has 3 integral digits and number 32 has 2 integral agi, Adding we have 5:integral digits and less one leaves 4 integral digits in our final answer. As we were only able to read three numbers on the scale we must add | zero to make the necessary 4 integral digits. Problems:' 14 x 35 x 8.4 Ans. 4120, 3.76 x .786x 185. Ans. 547. 27,5 x 12.5 x .27 Ans. 92.8 DIVISION Location of Decimal Point in Division by the Mental Survey Method In division we use the same scales, namely C and D, as were used in multiplication, only in reverse order. ; When dividing, the hair line of the indicator is placed over the number to be ‘divided into” (numerator) on the D scale; the slide is moved to the right or left so that the number we are “dividing by” {denominator) is located on the C scale will fall directly under the hair line. The answer will be found under the index of the C scale on the D scale. Example 1: 8 -- 4 = 2 ' Set the hair line of the indicator over 8 on the D scale, move the slide until 4 on the C scale is under the hair line; read the answer 2 under the left index on the D scale. (See Fig. 11.) ‘ ‘LAWTEENCE ENGINEERING SEFVICE, PERL, INDIANA. PAT. PEN | | | ol Lisantutnr Ras hitantntnnbtltarltittabfdstetaSetsttadlatetetdetedetatelsaeta toe gat turtasfaobytontoutauind anteast c hou 7 a Vy adebad t LM Wiis dite dad titatatentdtauthyadodertastubelbasdadebed vertetetdbetrtadutnnktaudesadenk Jalatpdot! Poadod Pk il * DIVIDING 8 BY 4 FIG. 11 Example 2: 845 ~ 25 = 33.8 From a mental survey we can sce we have practically 800 —- 20, giving us two places in the answer; thus we obtain 33.8 instead of 338 or 3.38;-or we can determine the decimal point in the following manner; 25 will go into 84 one digit and some left over; 25 will go into the left over amount and 5 another digit, thus giving 2 digits in our answer, so we have 33.8. The setting of the scale is shown in Fig. 13. Example 3: 6187 + 24.8 = 249 Here 24 will go into 61 one digit and give a remainder; this remainder and 8 can be divided by 24 making another digit in the answer and a remainder. Again this remainder and 7 can be divided by 24 making a third digit in the answer; thus we have three digits, 249 as the answer. Or we can say we have roughly 6000 ~ 24, giv- ing three numbers or digits in the answer. Division by the Integral Digit Method. (See page 21.) In division we use the same scales as are used in multiplication, * namely: the C and D scales because division is the reverse operation ‘to multiplication. ~ 16 0n D scale under the left index of the C scale. (See Fig. 12.) When dividing two numbers the hair line of the indicator is “placed over the. numerator or dividend on the D scale; the slide is moved to the right or left so that the denominator or divisor located on the C scale will fall directly under the hair line; read the answer - under the index of the C scale on the D scale. Example 1: 8 +4 = 2 ‘Set the hair line of the indicator over 8 on the D scale; move the slide unti! 4 on the C scale is under the hair line; read the answer 2 on the D scale under the left index of th C scale. (See Fig. 11.) Example 2: 48 + 3 = 16 Set the hair line of the indicator over 48 on the D scale, move the slide until 3 of the C scale is under the hair line, read the answer C8 RaedsaeaManndrsabinaatadaedotapbtaonebbbbtastacckrettt bettered tobanld Bubastbtontal par ctetvtahitanebianaduendndo Medd okedoted rerstentannibsattaaad adadubabattiatant boas Purea ple = raed ievshartuntuctontafrarTour@asted ha betasaSrastetietansdthtnuat sonatas { tude '.- ‘iategral digits in the answer is as follows: DIVIDING 48 BY 3 FIG. 12 Example 3: 845 -- 25 = 33.8 Set the hair line of the indicator over 845 on the D scale, move "the slide until 25 of the C scale is under the hair line, read the answer 33.8 on the D scale under the left index of the C scale. (See Fig. 13.) - t , e ed Sean feo Ue PAT. PEND. ft \ 12 f : cS 6 ‘ 4 i CU Faynatdsaabaamataatabunndhuttstibtedtstetettatatetetstebatitititeta tats tetatod ptats Beshada “c ‘aunetya AR, tu lat aPruifien eerstier B tnalt ei ult, Ha ea ie Wsiten DIVIDING 845 BY 25 FIG, 13 The user ‘will note that in the preceding examples the slide moved _ ~to the left.~ When this occurs the rule for finding the number of 4 —~ 1J0— ® IN DIVISION, WHEN THE SLIDE MOVES TO THE RIGHT, SUBTRACT THE NUMBER OF INTEGRAL DIGITS ° ~~! IN THE DIVISOR FROM THE NUMBER OF INTEGRAL DI- GITS IN THE DIVIDEND AND THEN ADD 1. THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE AN. SWER. Referring to example No. 1, the dividend 8 has one integral digit and the divisor has one integral digit; subtracting the one integral digit in the dividend leaves zero, but since our slide moves to the right, we add one making 0 + 1 = 1. Therefore, our answer will have one integral digit. Using example No. 2, the dividend 48 has two integral digits and the divisor 3 has one intergral digit. Subtracting one from two and then adding ! we obtain two integral digits in the answer. With example No. 3, the dividend 845 has three integra} digits ‘and the divisor 25 has two integral digits. Subtracting two from three and then adding | we obtain two integral digits in the answer. Problems: 72.6 + 1.83 Ans. 39.7 6187 -- 24.8 Ans. 249 5.493 — 37 Ans. .1484 813.6 ~~ 672 Ans. 1.21 Example 4: 763 -- .027 = 28,200 Set the hair line over 763 on the C scale, move the slide until 27 is under the hair line, read the number 282 on the D scale under the index of the C scale. The number 763 has three integral digits. - and .027 has minus one integral digit. Since we are dividing, we subtract the integral digit of the divisor from those in the dividend as stated in the rule. But here. we are subtracting a minus one from a plus three, which will give four as explained under “Subtraction of Integral Digits.’ Also the slide extended to.the right, so we must add 1, to the result just obtained by subtracting the integral digits, making 4 + 1 or 5 integral digits in our answer. Therefore, our answer 28200; Sag the two zeros to make the necessary five in- tegral digits. (See Fig. 14.) ¥ ee | RE ENGINEERING SERVICE, PERU, INDIANA. PAT. Pf = ¢ B pretee WW ; Cnty titi ft ‘tye YoumadaniePradabians Risiehisiiteitdtdtshda dled btittatnbatotettetalat deter titan vectra Jaduutustendustont c ry 2 3 6 7, Bg Uebaaaedtatataatietat Gtaheatin atic elite Litelhobteettetdabiades “4 it oa earTon nant : DIVIDING 763 BY .027 FIG. 4 Example 5: .0851 -- .0362 = 2.35 ‘ Set the hair line over 851 on the D scale, move the slide until 362 is under the hair line, read 235 on the D scale under the index. Here the dividend has a minus one integral digit and the divisor also has a minus one integral digit. Subtracting minus one from minus one we get zero, but the slide goes to the right so we must add one ‘to zero making one integral digit in the answer. Therefore, the — reading 235 becomes 2.35. (See Fig. 15.) oa JuamunRatatinututshianntonddaanhitbnbatnntinys whatotetatan Batotatotadaohatits Yaeitoovateg ’ © J © ta Satutecy » . be adetef De tntastetutahntodasduchintotunkad caTahidtond otal tstadtstatytdatsbaadhinad sasfiandnl ald ratind DIVIDING .0851 BY .0362 Fr. 15 Example 6: 3852 -- 725 = 5.31 Example 7: 143.7 +.9.36 = 15.35 In the examples above, 3852 has four integral digits while 725 described except that the slide will move to the left and the answer will be found under the right index of the C scale. When the slide moves to the left we have the following rule: EN DIVISION, WHEN _ JHE SLIDE MOVES TO THE LEFT, SUBTRACT THE NUM- ’ BER OF INTEGRAL DIGITS IN THE DIVISOR FROM THE NUMBER OF INTEGRAL DIGITS IN THE DIVIDEND‘ AND THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER. | In the example above, 3852 has four integral digits while 725 has three. Subtracting, we have one integral digit in the answer, giving 5.32. In example 7, 143.7 has three integral digits while 9.36 has one. Subtracting one from three gives two integral digits in the answer, so we have 15.36. Example 8: 26.8 -- .0652 = 411 Here 26.8 the dividend, has two integral digits and .0652, the divisor, has minus one integral digit. Subtracting minus one from two we obtain three, which is the number of integral digits in the answer. —_ }2 = ‘fn this problem the rule of subtraction of integral digits is used. as previously explained. Problems: 4873 —- 562 Ans. 8.67 93.14 — 472 Ans, .197 0568 -— .027 Ans. 2.1 6.35 —- 18.2 Ans. .349 Scale A. Scale A is divided into two halvés, called the left and the right half. (See Fig. 1.) The left half is divided into 9 unequal divisions, similar to scale D, and these divisions are also called UNIT divisions. Unit t is divided just as unit 2 is on the D scale, so the divisions will have the same value as did unit 2 of the D scale. Units 2, 3 and 4 are like unit 4 of the D scale and they have the same value as did unit 4 of the D scale. Unit 5 of the A scale is divided in ten major divisions only, so the third digit of any number must be estimated. This is also true of units 6, 7, 8 and 9. The right half of the A scale is divided as in the left half, there- fore the values of the divisions will be the same. As stated before, scales A and B are identical, as the readings on scale B will be the same as the reading on scale A. SQUARING A NUMBER BY THE MENTAL “SURVEY METHOD When we square a number we mean we are to multiply the num- ber by itself, thus 6? is 6 x 6. The small number 2 at the upper -ight of the figure means the number is to be squared. When squaring a number on the slide rule we use the D and A scales. The hair line of the'indicator is set over the number to be squared on the D scale and the answer is read under the hair line * on the A scale. Example 1: (2)? = 4 _.. Set the hair line of the indicator over 2 on the D scale and read “4 under the hair line on the A scale. © Mie Example 2: (4)? = 16 . Set the hair line of the indicator over 4 on the D scale and read inder the hair line on A scale. Example 3: (18)? = 324 i Make the gic ed has been explained in examples 1 and 2, The number 18 lies between 10 and 20; so 10? = 100 while 20° == 400, but since 18 is nearest to 20 the answer will be near 400; thus we have 324. DL eeererey pT TT TR TTT MTT 7! Ie n K Petesetetetetammetntaadinadatintiot Budtofisn] . HAIRLINE SQUARING 18 FIG. 166 Example 4: (36.2)? = 1310 The number 36.2 lies between 30 and 40; so 30? = 900 while 40°, = 1600, but since 36.2 is closer to 40 we will have four numbers, in the answer: thus 1310. Example 5:° (18.7)? = 66.7, The number 8.17 lies between 8 and 9; so 8 = 64 while. 9? = 81, but 8.17 is closer to 8 than 9 so we have two numbers in‘ the answer; thus 66.8 Problems: (92.4)? = 8540 (2.61)? = 6.81 - ( 158)? = 24,900 Squaring a Number by Integral Digit Method. (See page 21.) When we square a number we use the D and A scales. The hair line of the indicator is set over the number to be squared on the D scale and the answer is read under the hair line on the A scale. Example 1: (2)? = 4 Set the hairline over unit 2 on the D scale and read. 4 under — the hair line on the left half of the A scale. ~a4— Example 2: (3)? = 9 Set the hair line over 3 on the D scale and read 9 under the hair “fine on the left half of the A scale. . Example 3: (25)? = 625 In the above examples the answers fall in the left half of the A scale, so a rule for squaring in this case is: WHEN SQUARING A NUMBER, IF THE ANSWER FALLS IN THE LBFT HALF OF THE A SCALE MULTIPLY THE IN- . TEGRAL DIGITS IN THE NUMBER BY 2, AND THEN SUB. TRACT 1. THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER. , In example 3, the number 25 has two integral digits. Multiplying the number of ‘integral digits by 2 we obtain 4 but the answer falls in the left half so we must subtract one, making 4 — 1 = 3 integral digits in the answer. Therefore, 625 is the answer containing three integral digits. Example 4: (167.3)? = 27,900 The number 167.3 has three integral digits; multiplying this by 2 we obtain six integral digits, but it falls in the left half pbthe A scale- so we must subtract 1, leaving five integral digits in the answer. | Since we can only read 279 we add the necessary zeros, making 27900. We will now consider the squaring of a number when it falls in. the right half of the A scale. Example 5: (5)? = 25 Example 6: (6)? = 36 In the above examples the answer was found under the hair line in the right half of the A scale and in each case we had two times as many integral digits in the answer as in the number to be squared, From this we have the folowing Fue WHEN SQUAR- ING A NUMBER, IF THE ANSWER FALLS IN THE RIGHT HIALF OF THE A SCALE MULTIPLY THE INTEGRAL DIGITS. IN THE NUMBER BY 2 AND THIS WILL GIVE THE NUM- BER OF INTEGRAL DIGITS IN THE ANSWER. Example 7: (87.3)? = 7620 The number 87.3 has two integral digits. Multiplying by 2 we “~‘ebtain four which will be the number of integral digits in the answer. ~ OAS we are only able to read 76 on the A scale we add the two zeros “to make the necessary four digits. * , 42.93)? Ans. 1840 305.2)? Ans. 93100 753)? Ans. .567 613.4)? Ans. 376,000 Square Root of a Number by Mental Survey Method When we are going to find the square root of a number, we mean that we are to find a number which, when multiplied by itself, will equal the given number. Whenever the sign (V ) that square root is required. Before the square root can be found we must mark off the num- ber into groups of two digits each starting from the decimal point. For example, 3148 which is a whole number will be marked off into two groups with 4 and 8 together and 3 and 1 together in this manner: V 31/48]. Here the first group 31 has two digits. Another example 373 will be V3|73. with only one digit in the first group. Again in the numbers 538.42 and 6.471 they will be grouped as V 5|38.42 and v647]10. \ In the answer to the square root of a number there will be as many numbers in the answer as there are groups in the number itself. To find the square root of a number is the reverse operation of squaring therefore the A and D scales will be used again. IE the first group has one number, as shown by 3|73 or 5|38.42], set the hair line of the indicator over the number on the left half of the A scale and read the answer on the D scale. If the first group has two numbers as in 31|48. set the hair line of the indicator over 3148 on the right ea a the A scale and read the answer under the hair line on the scale. is placed over a number it means Example 1: V4 = 6.93 Since this group has two numbers, set the scale on 48 in the right half of the A scale and read 694 on the D scale. Here we have only one group, therefore we will have only one whole number in our answer; thus 6.94. Example 2: V 1/44 = 12 Here we have one number in the'first group, so we set 144 on the left half of the A scale and read the answer 12 under the hair line on the D scale. The number has two groups so we have two whole num- bers in our answer. e —16— Example 3: V17[64= 42.1 This number has two numbers in the first group so we set the indicator over 1764 on the right half of the A scale. This answer 421 is read under the hair line on the D scale. Our number has two groups, therefore we will have two whole numbers in our answer: thus 42.1. (See Fig. 17.) HAIRLINE FERING SERVICE, PERU, ING ? ranted asualvaatnsbodheol nytt vied SQUARE ROOT OF 17.64 FIG. 17 v13/82. = 37.2. v3/48]17. = 234. V 7.38 = 2.72 Square Root of a Number by the Integral Digit Method. (See page 21.). When we say we are going to find the square root of a given number, we mean that we are going to find an unknown number which. when multiplied by itself, will be equal to the given number. The sign (V ) when placed over a number signifies that the square root of that number is required. Before we can take the square root of a number, we must first divide the number into groups of two digits each, beginning at the decimal point and point off to the right and left as the case may be. For example, the whole number 4257, we begin at the decimal and point off to the left in groups of two digits, thus |42|57. When pointing off, we place the marks above the number so that they will not become confused with the decimal points. In this number we then have the first group as 42 and the second group as 57. In many cases we will have only one digit in the first group, as in the number 5|87|65. Here the first group is 5, the second group 87 and the third group 65. -—7— In a number containing whole digits and decimals, as 54257.4927, we mark off starting at the decimal point point right and left, thus ., . 5/42|57.49|27}. | Finding the square root is the reverse of squaring on the slide Rule. Therefore, the A and D scales are used again. When squaring a number we had two simple rules to learn, so when finding the square root we will also have two simple rules to learn. Ist. When the first group contains one digit, set the hair line over the number on the left half of the A scale and read the answer under the hair line on the D scale. . ; 2nd. When the first group contains two digits, set the hair line ’ over the number on the right half of the A scale and read the answer under the hair line on the D scale. In both cases there will be as many integral digits in the answers as there are whole groups in the numbers. | Example 1: Vv 1/44. = 12 The number 144 marked off into groups, giving us two groups before the decimal point; therefore, in answer we will have two in- tegral digits, thus 12. Example 2: V31[76.58 = 56.4 This number has two digits in the first group so we use the right ‘half of the A scale and read 564 on the D scale. But, since there are only two groups in the number before the decimal point we. will only ‘have two integral digits in the answer, thus 56.4. Example 3: Vv .71[30] = .844 Again the first group has two digits, but it also follows the deqj- mal point, so our first digit in the answer will follow the decimal point. Problems: V 367 = Ans. 19.16 V 57.426 = Ans. 7.58 V 1476.83 = Ans. 38.5 The “K” Scale. The K scale is divided. into three equal sections (See Figuré 1) which can be easily separated into a left third, a middle third and a /. ‘gight third. Upon examination of the scale the user will see that the divisions and markings are identical in all three thitds, Therefore an explanation of any one of these thirds will apply to the other two also.. With this in mind let us examine the left third. fl Example 4: (28)% = 21,900 The left third is divided into 9 unequal divisions, as has been found in all the other scales. Here again, each main division acts as a unit. Unit 1 has ten major divisions and each major division has five minor divisions just as was the case of Unit 2 of the D scale. Since the remaining units have divisions similar to other units of pre- ceding scales, the user will be able to determine the division values for his or herself. CUBING A NUMBER BY MENTAL SURVEY METHOD When we cube a number we mean we are to multiply the number. by itself three times, thus (2)® is 2X2X2. Fo find the cube we use the D and K scales. Set the hair line of the indicator over the number on the D scale and then read the answer under the hair line on the K scale. From Figure I we see that there are four ones or indexes. When the cube falls in the left third the answer will be between 1 and 10; if the cube falls in the middle third the answer will be between 10 and 100; if the cube falls in the right third the answer will be between 100 and 1000. This is repeated again, increasing each third making the left third read 1000 to 10,000, the middle third 10,000 to 100,000, etc, Example 1: (2)? = 8 ; Set the hair line of the indicator over 2 on the D scale and rea 8 on the K scale under the hair line. The answer falls in the left third where our numbering is from 1 to 10, therefore our answer is 8.. Example 2: (3)8 = 27 The answer falls in the middle third where the numbering is from 10 to 100, therefore we have 27. Example 3: (6)3 = 216 The answer falls in the right third where the numbering is from 100 and 1000, therefore we have 216. (See Fig. 18.) P. PEND.. Fatt ‘hoatunbutourbusityseuatiattinativsst 3 tovastarestessabessstunbnd a sectarsdeahest b er | : . MM HAIRLINE CUBING 6 ; a FIG. 18 : The answer falls in the middle third, which ranges from 10-100, or | q 10,000 to 100,000. Since 28 is close to 30'and the cube of 30 woyld |. : be in the 10,000 to 100,000 range, therefore 28? will fall in this range, —19— - ee Bas of a ae SA te Probiems: (8.45)? =~603 (12)3 = 1,730 (63.52)= 256.000 wubing a number by Integral Digit Method. (See page 21.) “When we cube a number we mean we are to multiply the number by itself three times, thus 6° is 6X6X6. To find the cube of a number on the slide rule we use the D and K scales, The hair line is set over the number to be cubed on the D scale and read the answer under the hair line on the K scale. When any number is to be cubed the answer may all in the left third, the middle third or the right third. Therefore, rules must be established to determine the number of integral digits in the answer. Rule No. 1. WHEN THE CUBE OF A NUMBER FALLS IN THE RIGHT THIRD, MULTIPLY THE INTEGRAL DIGITS IN THE NUM- BER BY 3 AND THIS WILL GIVE THE NUMBER OE INTE- GRAL DIGITS IN THE ANSWER. Example 1: (6)3 = 6X6X6 = 216 Set the hair line over the 6 on the D scale, read 216 under the hair line on the K scale. Following the above rule: number 6 has one integral digit, multiplying one by 3 we have three integral digits: re- ' quired for the,answer, thus 216. ‘(See Fig. 18.) Example 2: (81.5)? == 541,000 The number 81.5 has two integral digits, multiplying this by 3 we.will have six integral digits required in the answer. Since we can only read 54 on the K scale we add the four necessary zeros to make six integral digits. Example 3: (.572)* = .187 The number .572 has zero integral digits, multiplying by 3 we get zero. Thus we place the point in front of the number 186. Example 4: (.092)3 = .00078 The number .092 has minus one integral digit, multiplying minus _ one by 3 we get minus three integral digits required in the answer. As ’ we learned earlier in the instructions that minus three means three zeros, we place them before the number 76 read on the K scale. Therefore, our answer .00076, *3 giving two integral digits in the answer, thus 64. Rule No. 2. ; WHEN THE CUBE OF A NUMBER FALLS IN THE MID. DLE THIRD, MULTIPLY THE INTEGRAL DIGITS IN THE NUMBER BY 3 AND THEN SUBTRACT 1, THIS WILL GIVE THE NUMBER OF INTEGRAL DIGITS IN THE ANSWER, Example 1: (4)3 = 64 The number 4 has one integral digit, multiplying one by 3 we get 3, but the answer falls in the middle third so we subtract one from Example 2: (28.3)8 = 22,600 _ The number 28.3 has two integral digits, multiplying by 3 we get 6, but w