International projectionist (Nov-Dec 1933)

MATHEMATICS FOR THE PROJECTIONIST Gordon S. Mitchell IV. The Slide Rule DURING the many years since the science of mathematics has been developing, mathematicians have worked out tables and charts which present short cuts in performing desired mathematical operations. One of these short cuts is the log table; still another and a further development of the log table is the slide rule. The slide rule may be obtained in a great number of forms, in various sizes, and containing various tables for performing various operations. Slide rules designed for these special operations will not be considered in this lesson, but the projectionist should be entirely familiar with the principles governing the operation of the ordinary log slide rule, and, if possible, should have some facility in performing mathematical problems on it. On the slide rule, one scale moves along the other to permit various combinations of numbers on the two scales to be effected for the operations of multiplication and division. Before observing the relationships between these tvro scales, reference should be made first to the possibility of manipulating two identical measuring sticks, such as footrulers or meter sticks, for the performance of addition and subtraction. These are unit scales and begin with zero, instead of with one as on the Slide Rule scales. It should also be noted that the consecutive integers on the inch scale are equidistant, in contrast to the whole numbers on the Slide Rule scale, whose way with not a little momentum behind it. Can we duck this beating? I think "yes". But in order to duck it we will have to get down to brass tacks at once and resume where we left off five years ago when we first started flirting with that lady of false promises, Miss Apathy. Educational activities should be fostered by organizations — national, sectional, and local — and by this I do not mean to exclude the labor organizations. They should be in the very forefront of the movement, because, as Joseph Bliven remarked in these columns last month, they stand to get the most out of it. distances from each other decrease toward the right. Addition and Subtraction In order to add by measuring rather than by the usual method of counting, one might use one scale alone and a pair of dividers. For example, if the distance 3 on the unit scale were measured off with the dividers and laid next to 2, the total distance would be 5 on the scale, marked off by the right-hand leg of the dividers. However, to eliminate the use of dividers, two scales must be used. Now, if the beginning of the upper scale is placed directly over the 2 of the lower, 5 may be read on the lower. By setting off the distance 2 on the lower scale with the edge of the upper, 2 is added to any number on the upper scale and the sum read on the lower. However, when 2 is under 0, 3 is under 1, 4 under 2, 5 under 3, and so on for every successive number. Subtraction is merely the reverse of addition: using the addition problem outlined above which shows 2 + 3 = 5, the same numbers in the reverse order form a subtraction problem 5 — 3 = 2. It is easy to see how subtraction would be performed with these unit scales: If the 3 of the upper scale were placed over the 5 of the lower, 2 would appear as 2oo/oo loj 9o lay eo 1*^70 lfpt>0 log 50 lcg40 -■ kp-io- /.47 1&20- A30 Ig io\_ /. CO Figure 1 the remainder on the lower scale under the end (zero), of the upper, and the setting would look the same as for the addition of 2 plus 3. [15] 2.00 A9S A90 /■84 /■77 /.69 /■bO Addition is the building-up process and subtraction is the tearing-down process. By placing the 3 over the 5, the distance 3 is cut off of the distance 5, and the distance 2 remains projecting 4 5 6 7 8 9 Figure 2 out to the left. It should be clear that addition necessitates the measuring off of one of the distances first, in order to add it to the other on the second scale, the sum being read under the last distance. The slide rule proper, as it is used by engineers, is based upon the principal of logarithms, as was stated above. In Figure 1, we show the relation between the logarithm of a set of numbers from 10 to 100, the value of these logs being between 1 and 2. This relationship, in tabular form, follows: Log 10 = 1.00 Log 20=1.30 Log 30=1.47 ; Log 40=1.60 Log 50=1.69 Log 60=1.77 Log 70=1.84 Log 80=1.90 Log 90=1.95 Log 100=2.00 Two such scales, placed one above the other and arranged so that one may be moved in relation to the other, constitute a slide rule in its simplest form. (Fig. 2.) Multiplication Process In order to multiply with the slide rule, the top or movable scale is moved to the right. For instance, suppose we wish to multiply 3 by 2. The movable scale 1 2 3 4 5 6l 1 1 1 1 1 » 1 1 2 1 1 1 1 1 1 » 4-56769 Figure 3 is slid to the right until the left-hand index (the one on the movable scale), is directly over the 3 on the lower scale, as is shown in Figure 3. Reading directly under the 2 on the upper scale, we find the product of our multiplication which is 6. You will notice that should we want to multiply 3 by 3, we are able