Multiplication -- Method #3: D and CI Scales

And now we come to a real powerhouse of a method and the one you should reach for most often. While I'm getting ahead of myself here, know that division on a slide rule never leads to index-swapping. One of the two indices is guaranteed to be on-scale. (If division on a slide rule is  completely new to you, skip to the section dealing with it first, then come back here.)

Next, recall that multiplication is the same as dividing by the reciprocal.

And lastly, the CI scale (running in reverse compared to the C scale) represents reciprocals. That makes sense, doesn't it, for the negative of a logarithm is the logarithm of a reciprocal.

Put these altogether, then, and we have a guaranteed method. Let's try the same problems as before.

Example 3.1: Find the product of 3.5 and 2.7:
  1.     Set the cursor at 3.5 on the D scale.
  2.     Align 2.7 on the CI scale with the hairline.
  3.     Move the cursor to the right index.
  4.     Read the result under the hairline on the D scale: 9.45.
Remember: You can click on any image here for an enlarged view.

Cursor on D:3.5, align CI:2.7, result under index at D:9.45

The only tricky bit here is remembering that the CI scale reads in reverse. Many slide rules remind you of this by printing the numbers in red. On the rule I'm using here, you'll note that Pickett has placed a < sign next to reverse-layout numbers which suggests "read right to left."

Now let's try that product which gave us trouble with Method #1 earlier.

Example 3.2: Find the product of 4.2 and 6.5:
  1.     Set the cursor at 4.2 on the D scale.
  2.     Align 6.5 on the CI scale with the hairline.
  3.     Move the cursor to the left index.
  4.     Read the result under the hairline on the D scale: 27.30.

Cursor on D:4.2, align CI:6.5, result under index at D:27.30

Did you notice that in Example 3.1 the result ended up under the right index, but in Example 3.2 we found it under the left index? That's the beauty of division: one of the two indices is guaranteed to be on-scale.

Wrap-Up:
  • Advantages: There is no need for index-swapping, since either the left- or right-index will be on-scale. And the scales are all full-size meaning tired eyes can read the results easily. Moreover, virtually all but the simplest slide rules sport a CI scale.
  • Disadvantage: Really there is none, unless you want to count reading CI from right to left a disadvantage.
Next installment: Multiplication -- Method #4