Basic Methods: C, D and CI

In this installment we're going to take a peep into the business of chaining various operations. We've already seen some special cases in previous episodes, most notably when dealing with squares, cubes, square roots and cube roots. But for now let's try to lay out the methods in general. Well begin with some basic techniques requiring nothing more than the C and D scales, with a pinch of CI thrown in for good measure.

Now I'm not going to kid you: by far the most elegant and most accurate approach to chaining multiplications and divisions is by means of the folded scales CF and DF used alongside C and D. However, as those aren't necessarily available on every slide rule, it makes sense to learn how to grind things out with the elementary scales as well. Even if you have the folded scales at the ready, it still pays to learn the basic methods, since they might often provide a short-circuit in the middle of a more complex computation.

As a general rule of thumb, the goal will always be to minimize the number of slide movements since each introduces a bit of dither. And we certainly want to avoid the need for writing down intermediate results or executing number-transfers from one scale to another.

Let's begin by seeing how to form the product of more than two multiplicands.

Chained Products


Here's a fact of life: with only C and D at the ready, any product of n arguments is going to require n-1 slide movements. And that's if you're lucky enough to anticipate any index-swaps in the offing. Nonetheless, this is a good way to start.

Exercise 1.1: Compute 2.5 * 10.7 * 23.

Let's jump in with both feet and simply crank this out using canonical multiplication.
  1. Set the cursor at 2.5 on the D scale.
  2. Align the left index with the hairline.
  3. Move the cursor to 10.7 on the C scale.
  4. Align the left index with the hairline.
  5. Move the cursor to 23 on the C scale.
  6. Read the result under the hairline on the D scale: 615.
In summary, it looks like this on my Pickett 1006-ES. You can click the pictures to enlarge them if needed.

Cursor to D:2.5. left index to hairline, cursor to C:10.7
Left index to hairline, cursor to C:23, result at D:615

You'll note that it took two movements of the slide for three operands. However, everything went according to Hoyle, and there was no need for an index-swap (nothing went off-scale). That won't always be the case as we'll see shortly.

But also observe that were was no need to write down any intermediate result, let alone even read it or know what it was. In a sense, the cursor is your scratchpad memory.

And, of course, as usual you'll need to estimate the product at the end to arrive at the proper order of magnitude for the result.

For a little extra practice, see if you can compute 6! (6 factorial, or 1*2*3*4*5*6) using this approach several times in a row. You might need the right index in places, but it's pretty easy to see it coming.

Chained Mixed Operations


The Pickett Teaching Guide for Slide Rule Instruction by Maurice L. Hartung, (Pickett, Inc.: Santa Barbara, California, 1960) recommends performing mixed operations (ones involving products and quotients) in the order: divide, multiply, divide, multiply, etc. Let's see if that's a good rule with the following.

Exercise 2.1: Compute 45 / 3.8 * 75.
  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 75 on the C scale.
  4. Read the result under the hairline on the D scale: 890.
Graphically, we have:

Cursor to D:45, C:3.8 to hairline, cursor to C:75, result at D:890

Hey! That only took one slide movement to carry out two operations, so Dr. Hartung's advice appears apt. Let's try the same approach on a problem which is almost the same.

Exercise 2.2: Compute 45 / 3.8 * 95.

Go ahead; give the procedure from the previous problem a try...

Uh-oh! Step 3 is problematic; that 95 is off-scale to the right. Here are some work-arounds.

Method #1: Well, it might not be our favorite thing, but let's invoke an index-swap to get past the roadblock.
  1. Set the cursor to 45 on the D scale.
  2. Align 3.8 of the C scale with the hairline.
  3. Move the cursor to the left index on the C scale.
  4. Align the right index of the C scale with the hairline.
  5. Move the cursor to 95 on the C scale.
  6. Read the result under the hairline on the D scale: 1125.
In pictures:

Cursor to D:45, C:3.8 to hairline, cursor to left index, right index to hairline
Cursor to C:95, result at D:1125

Compared to Exercise 2.1, that was all far from elegant, but at least it got the job done. There were two slide movements, and one of them was an index-swap. I always hate those, since with a long-range travel I just know my aging eyes or haste are going to factor in some imprecision.

You might think that can be the best you can hope for with a slide rule sporting only the C and D scales. But keep reading.

Method #2: Same problem, but let's try a different tack. What if we commute things and work towards 45 * 95 / 3.8; might that help?
  1. Set the cursor to 45 on the D scale.
  2. Align the right index with the hairline. (Because the next operand, 95, is so far to the right, we could anticipate and avoid the index-swap).
  3. Move the cursor to 95 on the C scale.
  4. Align 3.8 of the C scale with the hairline.
  5. Move the cursor to the left index.
  6. Read the result under the hairline on the D scale: 1125.
Here's how she goes:

Cursor to D:45, right index to hairline, cursor to C:95

C:3.8 to hairline, cursor to left index, result at D:1125

That still took two slide movements, but you'll note they weren't so extreme, i.e., the travel was considerably shorter which can only help efficiency-wise. Perhaps most important, there was no index-swap.

Method #3: For our third approach, let's see if the old "divide by a reciprocal" trick with CI has any benefits to offer.
  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 the left index.
  4. Align 9.5 of the CI scale with the hairline.
  5. Move the cursor to the left index.
  6. Read the result under the hairline on the D scale: 1125. 
This is how it looks:

Cursor to D:45, C:3.8 to hairline, cursor to left index
CI:9.5 to hairline, cursor to left index, result at D:1125

Again, two slide movements were required, but just compare the photos from these three methods. In this last approach, the slide moved infinitesimally which implies you'll be burning far fewer calories if nothing else!

Okay, some of these chained operations were a little convoluted. Still, I keep coming back to: with even a simple slide rule (Rietz, Mannheim, etc.) you can do some pretty impressive things.

But if it's real elegance you crave, then lay your hands on a rule featuring the folded scales. Stop back in the next installment and you'll see how to cut all this slide movement in half.

Next installment: Chained Operations: Slick Approach with Folded Scales