Four Methods for Division

Finally, we get to division proper, even though I've cursorily alluded to it several times thus far. If you're comfortable with the previous techniques involving products, this'll be easy to add to your bag of tricks. I'll cover four methods for division here; you're likely to see all of them sooner or later, so why not master them now. They're illustrated by the following problem:

Exercise 1.1: Find 47.5 / 0.75.

Method #1: This is the "basic" or "standard" technique, which no doubt came first historically. It simply makes Napier's rule, log(x/y) = log(x) - log(y) physical, with x appearing on the D scale, and the C scale subtracting off y logarithmic units.
  1. Set the cursor at 47.5 (4.75) on the D scale.
  2. Align 0.75 (7.5) on the C scale with the hairline.
  3. Move the cursor to the right index.
  4. Read the result on D under the hairline: 63.3.
Of course, at the very end, we mentally adjusted for the order of magnitude, which I won't bother mentioning from now on. On my favorite slide rule, the Pickett 1006-N, it looks like this (which you can click to enlarge):

Cursor to D:4.75, C:7.5 to hairline, cursor to right index, result at D:6.33

The C and D scales are always available on every slide rule, making this a universal approach. Even better is that in step 3, one of the two indices is guaranteed to be on-scale, in this case the right index. You'll never need an index-swap with Method #1.

Method #2: The basic approach, above, can also be used on the folded CF and DF scales. That'll be important to remember when we get to the topic of chained operations. Let's give it a run-through.
  1. Set the cursor at 47.5 (4.75) on the DF scale.
  2. Align 0.75 (7.5) on the CF scale under the hairline.
  3. Move the cursor to the right index (on C).
  4. Read the result under the hairline on D: 63.3.
Observe that we made a scale-transfer in step 3; the right index is on C and we read the result on D. It looks something like this:

Cursor to DF:4.75, CF:7.5 to hairline, cursor to right index, result at D:6.33

But further note that the scale-transfer is not absolutely necessary. With the slide rule still set up, instead of step 3, align the cursor with 1 on the CF scale, which behaves as sort of a pseudo-index. The result is then under the hairline on the DF scale. But bear in mind, should you decide to use the pseudo-cursor, a rather hardcore index-swap will be required at times when the 1 runs off-scale. For example, try DF:5 divided by CF:2 and see for yourself.

Method #3: This should seem pretty familiar by now: we're going to use the A and the B scales for the same problem.
  1. Set the cursor at 47.5 (4.75, and either one) on the A scale.
  2. Align 0.75 (7.5, again either one) on the B scale under the hairline.
  3. Move the cursor to the right index.
  4. Read the result under the hairline on A: 63.3.
Apart from the strain on the old peepers, everything is pretty straightforward.

Cursor to A:4.75, B:7.5 to hairline, cursor to right index, result at A:63.3

Method #4: You'll recall from the previous section that the CI scale runs in reverse to provide reciprocals. So, here we'll think of x/y as x times 1/y, and proceed as though this were a multiplication problem.
  1. Set the cursor at 47.5 (4.75) on the D scale.
  2. Align the left index with the hairline.
  3. Move the cursor to 0.75 (7.5) on the CI scale.
  4. Read the result under the hairline on D: 63.3.
And here's how it looks:

Align left index with D:4.75, cursor to CI:7.5, result at D:6.33

In step 2, we used the left index, since the right index would have put the CI:7.5 off-scale. However, supposing your slide rule sports the folded CIF scale, that wouldn't be a problem anyway; just do a scale transfer and conclude on the DF scale.

As a side-note, I think you're starting to see why I so adore the Pickett 1006-ES. Note only is it the daintiest of pocket rules and slides smooth as silk, but with the C, D, CI, DI, CF, DF, CIF, A and B scales (among others) the sky is the limit. With just a bit of practice one can find a magnificent array of products and quotients with very little movement of the slide or cursor.

Wrap-Up: Of these approaches, Method #2 presupposes you have the folded scales, which you won't find on the typical Rietz type layout.

Method #3 is the most taxing on the eyes, since the graduations of the A and B scales are rather fine (especially on a 6" rule). However, the method is still worth learning, since it leads very naturally to the notion of "special quotients," i.e., those involving squares and square roots.

And, Method #4 (multiplying by a reciprocal) may seem a bit contrived, however it'll prove its merit when we finally get to chained operations.

So unless you have some ulterior motive in mind, Method #1 is the way to go, at least for garden variety quotients.

Next installment: Special Quotients