Before we get into division proper, it'll be wise to take at a look at reciprocals for a moment. After all, division is defined as multiplication by a reciprocal.
There are at least three ways to find the reciprocal of a number. Let's investigate each by means of the following problem.
Exercise 1.1: Find the reciprocal of 3.5.
Method #1: The approach that would come to mind first with most people is simply to divide 1 by 3.5, and that's what we'll do here. Of course, this presupposes you know how to divide on a slide rule, which is in fact the topic of the next section. So, strictly speaking, we're putting the cart before the horse.
However, since the method is so straightforward let's just go with it. Best of all, this requires nothing fancy, just the C and D scales, so is available on even the simplest slide rule.
- Set the cursor at 1 on D, at the left.
- Align the slide with 3.5 on the C scale under the hairline.
- Slide the cursor to the right index.
- Read the result under the hairline on D: 0.286.
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| C:3.5 aligned with D:1, cursor to right index, result at D:0.286 |
Like I mentioned, we'll find out more about division in the next section. But for now, as a bit of a hint, take at look at D:2 and see what lies directly above it on the C scale: 7. And 2/7 is also 0.286. Starting to see the pattern?
Method #2: This next approach might be considered the standard way of handling reciprocals. It uses the CI scale which, while somewhat newer as things go in the history of slide rules, is commonly available on just about any unit nowadays.
The CI scale is identical to the C scale...if you hold it up to a mirror! Now think about this for a moment. According to another of Napier's rules, log(1/x) = -log(x). Thus, if the CI scale runs in reverse, measuring logs in the negative direction, then it must represent logs of the reciprocal. Let's give it a whirl.
- Set the cursor at 3.5 on the C scale.
- Read the result under the hairline on the CI scale: 0.286.
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| Cursor at C:3.5, result under hairline at CI:0.286 |
Recall that reciprocals are a two-way street, you could have just as easily aligned the cursor with 3.5 on the CI scale and read the result off of the C scale as 0.286. Or if you close the rule completely so that C and D are aligned, you can also find the desired reciprocal on the D scale.
But wait, there's more! This great little Pickett 1006-ES I'm using also sports a DI scale. So, one could additionally fetch the result on either D or DI as well.
And, remarkably, this amazing slide rule also works in a CIF scale which can be used in tandem with the CF scale as yet another way to find 1/x. In short, Method #2 actually provides at five different paths to finding reciprocals, depending where you want the result to land. Sweet!
Method #3: Here's a whimsical approach that occurred to me earlier tonight. I would hardly call it the most practical way for finding a reciprocal, but it serves as a reminder that there's all sorts of internal symmetry and beauty within a slide rule.
The idea is this: 1/x = x/(x^2). And we know from the previous section that it's trivial to find squares with the help of the A scale. Let's try it out.
- Set the cursor at the leftmost 3.5 on the A scale.
- Align 3.5 on the C scale under the hairline.
- Move the cursor to the right index.
- Read the result under the hairline on the A scale: 0.286.
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| Cursor on A:3.5, C:3.5 under hairline, cursor to right index, result at A:0.286 |
You'll note in step 2, above, that 3.5 on the C scale is actually the square of 3.5, since its distance from the origin is twice that of 3.5 on the A scale. In other words, in Method #3, we're actually doing the calculation on A, with a brief scale-transfer to find the square of the divisor on C.
And in step 1, it really doesn't matter if you employ the rightmost 3.5; you'll just wind up using the left index in step 3. You'll recall from Multiplication by Squares, that your choice makes no never-mind when dealing with squares; it's square roots where you have to count digits and choose properly.
As a stand-alone computation, this approach has little to commend it. But who knows? Maybe in a chained operation it would help cut down on some index-swapping. More important, however, is that this crazy little diversion is wonderful training in thinking about how the scales are laid out.
Wrap-Up: Pretty clearly, Method #2 using the CI scale is the cleanest approach in most situations. I'm a lifelong fan of laziness, and not needing to move the slide has definite appeal..what is that, one or two calories saved?
So there you have it. With a bit of a foreshadowing of what's to come, we now know a variety of ways to find reciprocals. With that, it must be time to examine how to divide two numbers on a slide rule. Proceed to the next section.
Next installment: Four Methods for Division