Multiplication -- Method #4: CF and DF scales

I like to consider the folded scales passing lanes. Just when you think you're about to be slowed down by an index-swap, they provide a speedy alternate route which avoids unnecessary slide movement.

Though they've been "folded" at π (i.e., commencing at π, then wrapping around), there's really no reason you can't use them in lieu of C and D for ordinary multiplication, with the usual caveat of a possible index-swap needed.

Note: I'm told a very few slide rules fold the scales at square root of 10, which is theoretically where it should happen. After all, the midpoint of a logarithmic scale from 1 to 10 lies at this number. However, π and square root of ten are close neighbors (about 3.1416 and 3.1623, respectively). The advantage of using π is that this opens all sorts of possibilities for doing rapid computations involving the circumference of and area within a circle, among other things.

The very slight disadvantage is even crazy to think about: if you're computing with a number which lies between π and square root of 10, then you may have to do an index-swap with Method #1 multiplication. Yeah, really, like that happens all the time...
 
Back to the action. Let's try our usual problem, but this time using nothing more than the DF and CF scales. Observe that the new index used here now falls inside the CF scale.

Example 4.1: Find the product of 3.5 and 2.7:
  1.     Set the cursor at 3.5 on the DF scale.
  2.     Align the index (1 on the CF scale) with the hairline.
  3.     Move the cursor to 2.7 on the CF scale.
  4.     Read the result under the hairline on the DF scale: 9.45.
Remember: You can click on any image here for an enlarged view. 


Cursor on DF:3.5, align CF:1, result over CF:2.7 at DF:9.45

Still works even with the shift! But there's a better reason to become familiar with the folded scales. In particular, whilst doing multiplication with, say, the C and the D scales, if you're about to fall off the end, you can generally transfer things to the CF or DF and conclude the computation without an index-swap. I'll call this a scale-transfer. Let's try that troublesome Example 1.2 from Method #1 yet again and see how it all works.

Example 4.2: Find the product of 4.2 and 6.5:
  1.     Set the cursor at 4.2 on the D scale.
  2.     Align the left index with the hairline.
  3.     Move the cursor to 6.5 on the CF scale.
  4.     Read the result under the hairline on the DF scale: 27.30.

Left index on D:4.2, cursor on CF:6.5, result at DF:27.30

In Step (3), it looked like we were in trouble using the original method: 6.5 is way off scale to the right. So instead we slide the cursor to 6.5 on the CF scale and read the result off of DF. So the term scale-transfer is well justified.

Wrap-Up: The thing to note is that we didn't have to back up to restart things with the right index, but just forged ahead with the folded scales. Their utility really comes to the forefront when carrying out chained operations, of which more later.
  • Advantages: There is no need for index-swapping, since doing a scale-transfer generally keeps things near the inside. The scales are all full-size making them easy to read.
  • Disadvantage: Simpler slide rules (like those of the Mannheim and Rietz designs) do not have the folded scales.
You might note that there is a bijection between the C and CF, and D and DF scales. Thus, in Example 4.2, we could have just as easily started on the DF and CF scales then performed a scale-transfer to D and C to acquire the final result.

And with that, we've concluded our survey of basic multiplication techniques on a slide rule. Again, I suggest it's worth knowing all four methods, if for no other reason than that you'll more readily understand (and appreciate!) how a slide rule is constructed.

Next installment: Square Roots and Squares