However, even if your slide was lined up with the scales on the body, but otherwise frozen in place, you could use your slide rule as a lookup table for a large number of formulas.
Some of them are listed in Tables 2 and 3. For example, if you wish to compute the expression move the hairline over on the CF or DF scale, and read the result on the W scale.
More generally, if you choose a number on a scale corresponding to the function as listed in Table 1 , and you read the corresponding number on a scale corresponding to the function , then. Note that is not the number under the hairline on the C scale, unless you choose to start on that scale! As the tables clearly indicate, if you move the hairline over any number on any scale at all, and read the number on the same scale right under the hairline, you'll get that very same number back!
Two Variables Of course the number of possibilities is vastly increased by allowing the slide to move. Scales 1 and 3 are on the body, scale 2 is on the slide. PLUS: Select u on scale 1 on the body , align it with the index of scale 2 on the slide , move the hairline to v on scale 2, and read the result on scale 3 on the body , underneath the hairline.
For example if the scales involved are D , C , and D , the result would be the product, uv. MINUS: Select u on scale 1, align it with v on scale 2 on the slide, move the hairline to the index of scale 2, and read the result on scale 3 on the body, underneath the hairline. For example, if the scales involved are again D , C , and D , the result is the quotient,.
What happens if we use other scales? Assuming a very hypothetical slide rule that has all the scales listed above both on the body and on the slide, these two procedures let you evaluate 3, different expressions in 4, different ways.
Six examples are given in Table 4. Click here to see a similarly organized pdf file of several hundred pages showing all the possibilities. In general, if is the function corresponding to scale 1 again, as listed in Table 1 , the function corresponding to scale 2, and the function corresponding to scale 3, then the result that you read on scale 3 is.
The symbol indicates whether to use the plus or the minus procedure. The first three rows of Table 4 show the most common operations on a slide rule: product, quotient, and power. The last three rows show less common formulas that can be evaluated.
The first number in that row, , indicates the entry in the pdf table , 26 means it is the 26th distinct formula in the table, and 2 means it's the second way to evaluate this particular formula. These numbers are not important for the example, but they illustrate the organization of the pdf table. Caveats apply even more so than to the one variable Table 2 and 3 above. The variables have to be in certain ranges, and you may have to be judicious about which variant of the relevant scale you use to read your result.
Of course, slide rule manuals do not list thousands of formulas. They describe basic principles and then people can figure out how to use slide rules to best advantage for their particular applications. There are more pedestrian ways to compute but if you have to evaluate such expressions many times you'll find the shortcut eventually. Once you have it you can impress your friends and coworkers! The last example in Table 4 requires an LL scale on the slide.
When I went to high school our work horse slide rule was the Aristo Scholar One version of it has a body and cursor with one side, but a slide with two sides. The back of the slide shows several LL scales. So prior to doing this calculation you need to turn the slide around. This gives you a very strange slide rule without a C scale. For years I have wondered for what kind of application one would want to turn the slide on the Aristo Scholar, and after writing this web page I know!
Three Variables Suppose we consider a variant of the PLUS procedure where instead of the index we use a number on a fourth scale. Thus we start again by putting the hairline above the number u on scale 1. Then we move the number v on scale 2 underneath the hairline. Next we move the hairline above the number w on scale 3. Finally we read the result on scale 4 underneath the hairline. Scales 1 and 4 are on the body, scales 2 and 3 on the slide. With the 13 scales assumed here, there are 24, distinct such expressions, filling 2, printed pages that you can view or download here.
The four columns following the mathematical expression give the scales 1, 2, 3, and 4 being used. Sophisticated Multiplication and Division Sophisticated Multiplication sounds like an oxymoron, but it isn't in slide rule lore. We can multiply and divide using the C and D scales, and so in particular we can multiply with and compute reciprocals.
The purpose of these additional scales is to make multiplication and division fast and easy by minimizing the number of times and the distances that the slide and cursor have to be moved, particularly when doing repeated division and multiplication.
Try it, and you'll see that it is especially convenient if multiplications and divisions alternate. If you have a sequence of multiplications only you can replace some of them with a division by the reciprocal of the relevant factor, using the CI and DI scales.
It so happens that is close to that square root and works almost as well. In addition however, it makes it possible to multiply or divide by without any slide movement at all. At some stage in the past someone had the quite brilliant idea to approximate the square root of 10 by. Quadratic Equations As discussed above, one thing slide rules can do that calculators can't is create tables. The small tick marks between "10" and "20" divide the area between them into ten parts, so each small tick mark in that area stands for 1.
Thus, we can see that the 4 stands under 10 and two tick marks, or From 20 to 50, there are only five tick marks between each number and the number that is 10 greater. So each tick mark stands for 2. Thus, we see that 3 times 7 is 21, since the 7 is under a point about halfway between the 20 and the first tick mark following it. On the other hand, that 3 times 8 is 24 is easy to see, as that falls under the second tick mark after And, since 3 times 9 is 27, again we see that the 9 is between the tick marks that represent 26 and Noting how the scales on a slide rule simply repeat themselves, one can put the 10 instead of the 1 under the number by which one wishes to multiply to see the other possible products.
This lets one use the C and D scales, instead of the A and B scales, which are on a smaller scale, to multiply, giving more accuracy. Having seen how to do basic multiplication on a slide rule, we can now understand the basic scales on a slide rule, as shown in the first image above. This is not terribly complicated in itself; between the numbers on a slide rule, there are tick marks large and small, laid out in much the same way as they would be on a ruler or on the tuning scale of a radio.
However, because slide rules multiply instead of adding, their scales aren't uniform like that of a ruler. To make it possible to read numbers with precision, the tick marks are kept close together, and so along the length of the rule, their scheme changes. Thus, in the illustration above, the space between the numbers 1 and 2 is first divided into halves by one large tick mark, then into tenths by several medium tick marks, and the tenths are themselves divided into five parts by small tick marks.
So on this part of the slide rule, the space between two adjacent tick marks represents one-fiftieth of that between two numbers on the scale. Between the numbers 2 and 5, the space between two consecutive digits is first divided into halves by a large tick mark, then into tenths by medium tick marks, and each tenth is divided in half by a small tick mark.
So here, the space between two adjacent tick marks is one-twentieth of that between two numbers on the scale. And between 5 and 9, the scale looks like the centimeter scale on a typical ruler; the space between two numbers is divided into halves by a large tick mark, and then each half is divided into five parts by a small tick mark.
Here, the space between two adjacent tick marks is one-tenth of that between two numbers on the scale. This is how the C scale would be marked on a typical 5-inch slide rule. On a inch slide rule, space for twice as many tick marks would be available, and so the layout would be different. Thus, the difficulty really is that it's necessary to pay attention to how the tick marks are laid out on the part of the slide rule on which one wishes to find a number at any given time.
If you multiply a number by itself, the result will be twice the distance from the index - the 1 mark - as the original number was. So if you have two slide rule scales, one twice as big as the other, if their indexes are aligned, each number on the scale that is twice as big corresponds to the location of its square on the smaller scale. Thus, the A and B scales, at half the side, with the scales repeated twice, aren't just for beginning slide rule users.
Rather, they let people work with squares and square roots. Similarly, the K scale is at one third of the scale of the C and D scales, and so it is used for cubes and cube roots.
Understanding How a Slide Rule Works. Go back. Raw Number Log Scale base Linear Logarithmic Scale base
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