Slick Approach with Folded Scales

In the previous installment, we saw that it's possible to chain a sequence of multiplications and divisions without needing to jot down intermediate results. However, from time to time, we also noticed that some rather tedious adjustments were needed to keep from going off-scale, things like index-swaps or commuting the operands. Moreover, there really were way too many slide movements in general.

We can lick all of those problems by using a slide rule which also includes the folded scales CF, DF and CIF. My little Pickett 1006-ES sports those, so that's what I'll continue using in the examples.

Of all the scales on a slide rule, the folded scales (simple as they are) perhaps provide the greatest convenience. The essential idea is that whenever an index has gone off-scale, you can do a scale-transfer to the folded scales and continue on as though nothing untoward has happened. And if that eventually takes you beyond the left or right limits, you can scale-transfer back again. Voilà! No index swapping or excessive slide movement required!

Let's try the same problems from the previous installment just to see how slick things turn out now.

Exercise 1.1: Compute 2.5 * 10.7 * 23.

If you check your notes with how we performed this chained operation last time, you'll note that it took two slide movements. Get ready to save some energy...
  1. Set the cursor at 2.5 on the D scale.
  2. Align 10.7 on the CI scale with the hairline.
  3. Move the cursor to 23 on the CF scale.
  4. Read the result under the hairline on the DF scale: 615.
Click the following photo to enlarge it and see how simple it all turns out.

Cursor to D:25, align CI:10.7 with hairline, cursor to CF:23, result at 615

Notice in step 2 we did the "divide by a reciprocal" thing to multiply by 10.7. Then immediately thereafter, searching for 23 on C revealed that it's off-scale. So instead, we find it on CF, reading the final result on DF.

As usual, at the end it was easy to estimate the order of magnitude; just think 2 times 10 times 20 = 400 and you'll know the result must be 615, not 61.5 or 6150.

This is really cool, don't you agree? Just one thing: when you do a scale transfer, do it completely. In other words, originally we were measuring with C along D. Then we migrated the pair simultaneously so CF was measuring along DF. As a general rule, never measure C against DF or CF against D (unless you have some sort of computation involving π in mind.)

For a little further practice, go ahead and find 6! (6 factorial) once more. This time you'll be able to do it with two slide movements only.

Exercise 2.1: Compute 45 / 3.8 * 75.

Here we have a mix of products and quotients, but it's just as easy.
  1. Set the cursor at 45 on the D scale.
  2. Align 3.8 on the C scale with the hairline.
  3. Move the cursor to 75 on the CF scale.
  4. Read the result under the hairline on the DF scale: 890.
Graphically we have:

Cursor to D:45, align C:3.8 with hairline, cursor to CF:75, result at DF:890

You'll note in this example, you had a choice of C or CF for that 75 in step 3. You won't always be so lucky as the next example shows. I had you use CF just for practice.

Example 2.2: Compute 45 / 3.8 * 95.

In this case, the 95 of C is off-scale to the right, so just switch over to CF and carry on.
  1. Set the cursor at 45 on the D scale.
  2. Align 3.8 of the C scale with the hairline.
  3. Move the cursor to 95 on the CF scale. (You've no choice now!)
  4. Read the result under the hairline on the DF scale: 1125.
Here's what it looks like.

Cursor at D:45, align C:3.8 with hairline, cursor to CF:95, result at DF:1125

But there's no reason to stop at three operands. We can extend this process as long as we want.

Exercise 3.1: Compute (12.5 * 37 * 0.18) / (45.2 * 11).

One of the nice things about using the folded scales is that there's no need to commute any operands for convenience sake. In this problem we'll simply multiply the three operands of the numerator in order, then divide by 45.2 and divide again by 11. Sweet!
  1. Set the cursor at 12.5 on the D scale.
  2. Align 37 of the CI scale with the hairline.
  3. Move the cursor to 0.18 on the CF scale. (The one on C is off-scale to the left).
  4. Align 45.2 of the CF scale with the hairline.
  5. Move the cursor to 11 on the CIF scale.
  6. Read the result under the hairline on the DF scale: 0.167.
In pictures, the steps look like this:

Cursor at D:12.5, align CI:37 with hairline, cursor to CF:0.18
Align 45.2 with hairline, cursor to CIF:11, result at DF:0.167

By this time, you should be pretty skilled at using the inverse scales. In step 2, we're dividing by a reciprocal with inverse scale CI. But in step 5 we're multiplying by a reciprocal with CIF (i.e., an actual division).

Here are a couple comments to wrap things up. First off, be sure to remember that scale-transfers are bidirectional. You can go from C/D to CF/DF or vice versa with equal ease. Anytime you're faced with an off-scale number, just switch over.

Second, Professor Herning (of whom I've referred to frequently before) has investigated the use of the folded scales for chained operations with some perspicacity. He arrived at a particularly useful rule-of-thumb: any time more than half of the slide is sticking out of the stator in chained operations, make the switch mentioned in the preceding paragraph and carry on with the alternate scales. It really works!

Be sure to spend some time with Professor Herning's excellent YouTube episode which treats chained operations with the folded scales in great detail:


Were I a scientist or engineer faced with a daily deluge of computations, I'd want to be sure the slide rule I used possessed C, D, CI, CF, DF, CIF, A, B, K, S, T, ST and L at the minimum. Just so happens my favorite pocket rule, the Pickett 1006-ES has those, with a DI scale thrown in for good measure.

Next installment: Degrees, Radians and Arc Length