Slide Rule - Proportion, Percentage, Squares And Square Roots (1944)

e a study of the standard slide rule shows that of all scales on the rule the c and e scales are the ones most commonly used for multiplication and division let us review these procedures using a simplified rule with enlarged figures to multiply 2.5 by 16 for example first find 25 on D and to it slide the index of C move the hairline to 16 on C and read the answer 40 under the hairline on D the ciphers and decimals required by the problem are of course placed mentally the procedure may be diagrammed like this 2.5 * 16 equal 40 to review an example in division say 30 ided 4 first find three on D and to it slide four on C move the hairline to the index and read the answer 7.5 on D again the ciphers and decimals required are added mentally division procedure may be diagrammed like this 30 divided by 4 equals 7.5 when skill has been acquired in multiplication and division problems in proportion may also be solved with the C and D scales let us take the simple proportion 4 is to two as 6 is to X the proportion may also be stated like this a form more convenient when using the slide rule to find X first set the rule to match the known quantities above and below the line in this case 4/ 2 find two on D and to it slide four on C find six on C opposite 6 read X as three on D with this problem X is to two as six is to three the known quantities above and below the line are 6 over 3 so we find three on D and to it slide six on C find two on D and read X as four on C with this as with any setting of the rule all coinciding readings are in the same ratio to one another thus we may read four is to two as six is to three and as eight is to four as 10 is to five as three is to 1.5 as two is to one also using larger numbers in which positions must be estimated four is to two as 51.4 is to 25.7 and as 349 o is to 1745 in proportion as in multiplication and division the ciphers and decimal points must be placed mentally suppose now we wish to extend the equation to 4 is to 2 as 12 is to X we see that 12 is past the body of the rule if the ccale repeated itself we could read as 12 is to six actually the C scale can be repeated in effect by placing the hairline on the right index and then sliding the left index to it in shifting from one index to the other the ratio remains unchanged in this case the rule is still set in the ratio of 4 to2 we can read 4 is to two as 12 is to 6 as 15 is to 7.5 as 20 is to 10 now to take a typical problem in proportion here is a post 30 ft High casting a shadow 5 ft long the tower casts a shadow 54.5 ft long how high is the tower the problem may be stated in terms of proportion 5 is to 30 as 54.5 is to X to three on D Slide Five on C push the hairline to 545 on C under the hairline read the answer 327 on D the height of the tower is 327 ft suppose the problem is to convert inches into millimeters and millimet into in since 1 in equals 25 and 410 mm we write 1 is to 25.4 as 1.6 is to X as Y is to 43.2 as 8.5 is to Z to 254 on D set one on C move the hairline to 16 on C and read X as 406 on D move the hairline to 432 on D and read y as 7 on C now since 85 is off the body we must shift to the other index set the hairline to 85 on C and read Z as 216 on D again placing decimal points in terms of the problem we have 1 is to 25.4 4 as 1.6 is to 40.6 as 1.7 is to 43.2 as 8.5 is to 216 problems in percentage may also be worked by proportion of 285,000 men 91,000 are skilled workmen 176,000 unskilled and 18,000 unemployed what is the percentage in each group 285,000 is 100% find the percentages of X Y and Z stating the problem in terms of proportion 285,000 is to as 91,000 is to X as 176,000 000 is to Y as 18,000 is to Z to one on D set 285 on C under 91 on C read X as 319 since 176 is beyond the body shift to the other index and under 176 on C read y as 618 under 18 on C read Z as 63 as a final step add the decimal points notice that with any setting of the rule all coinciding readings are in the same ratio with this setting 285 is to 100 as 91 is to 31.9 and with a change of index which leaves the ratio unchanged as 176 is to 61.8 as 18 is to 6.3 with this setting four is to two as six is to three as8 is to four and so on it is this feature of the C and D scales which permits the ready solution of problems in proportion and percentage the C and D scales together with two others the A and B scales are used to obtain squares and square roots the A and B scales are made as though the C and D scales were contracted to half their length and repeated on the A and B scales there is not room for as many marks as on the C and D scales Within These brackets each smallest division is read as a two for example 100 102 104 106 100 102 104 106 Within These brackets each smallest division is read as a five for example 200 205 210 215 200 205 210 215 Within These brackets each smallest division is read as a 10 for example 500 510 520 500 510 520 in obtaining squares and square roots use the B and C scales or the A and D scales to square a number eight for instance place the hairline on eight on D follow the hairline up to the a scale and under the hairline read the square 64 on a conversely the square root of 64 is 8 the procedure for obtaining squares always takes this form the procedure for obtaining square roots always takes this form notice that to obtain the square roots of singled digigit numbers the left half of the A and B scales is used of two digigit numbers the right half if we try to obtain the square root of four on the right half we read 632 which of course cannot be the answer if we try 40 on the left half we read two which of course is wrong remember in finding square roots use the left half for single digits the right half for two digits with numbers of more than two digits mark off in pairs from right to left here a single digit remains to the left of the mark use the left side with this number two digits remain to the left of the mark use the right half with this number use the left half if the number contains decimals mark off in pairs to the left from the decimal point in this case since two digits main use the right side if there is no figure ahead of the decimal point mark off in pairs to the right from the decimal point until a figure which is not a cipher is included in a pair if the first figure in this pair which you have marked off is a cipher use the left side but if the first figure in the marked off pair is not a cipher use the right side in any mathematical problem it is of the utmost importance that the decimal point be correctly placed in the answer in obtaining squares and square roots there is a simple means for placing the decimal point take this problem for example we read the answer 1875 where does the decimal point Belong One under the three the eight under the 52 the seven under the 51 the five is left place it to the right of the decimal point and we have the answer 187.5 with this problem place the decimal point for the answer place a cipher under the pair of ciphers the one under the next pair then the 875 the answer is 0.1875 if the decimal point were here in the problem place the decimal point then the one under the pair then the 875 the answer is 1875 if the problem is to square a a number we merely reverse the procedure in order to locate the decimal point correctly in the answer reviewing squares and square roots read up to square a number read down to obtain the square root determine whether to use the left or the right half Loc the decimal point by this method as an example of a practical problem involving squares and square roots a piece of land is in the shape of a right angle triangle one side measures 150 ft another 65 ft what is the dimension of the long side x x in this example is the hypotenuse of a right angle triangle we know that the square of the hypotenuse of a right angle triangle is equal to the sum of the squares of the other two sides in other words x^2 = 150 2 + 65 2 then x equal the square root of the sum of 150 SAR and 65 SAR we Square 150 and obtain 22500 Square 65 and put down 4,225 adding these we get 26725 the square root of this figure will be the hypotenuse X by marking off we see we must use the left side placing the hairline on the estimated position for 2 6 725 on a we read 16 32 on D by lining up the figures thus we find that x = 63.2 with any right angle triangle where two sides are known the finding of the unknown side involves only the simple process of obtaining squares and square roots careful practice is essential to using the slide rule with speed and accuracy but facil and D scales makes possible the ready solution of problems in multiplication division proportion and percentage using the A and B scales with the C and D scales it is possible to obtain quickly both squares and square roots and these are only a few out of the wide variety of problems that can be worked with the standard slide rule for