The easiest way to answer that is by looking at two examples which illustrate, once again, that there's usually more than one path to the solution of a problem on a slide rule. I'm telling you, the internal harmony of logarithms made physical is ever there for those who view mathematics as one of the fine arts.
You'll observe that all of these examples depend upon a scale-transfer, back and forth between A/B and D/C, with CI thrown in as icing on the cake.
Exercise 1.1: Find the product of 2 and sqrt(3):
Method #1: Let's just jump in and compute this product by what would come to mind first. In words, we'll use the full-length scale D to represent 2, and the half-length scale B for sqrt(3). (Recall Napier's rule that half the log of an argument is the log of the square root of that argument). Specifically:
- Set the cursor at 2 on the D scale.
- Align the left index with the hairline.
- Move the cursor to 3 on the B scale.
- Read the result under the hairline on the D scale: 3.46.
Remember: You can click on any image here for an enlarged view.
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| Left index on D:2, cursor on B:3, result at D:3.46 |
Method #2: In some computations with some numbers, you'll find it beneficial to rearrange the order of the multiplication (remembering that that operation is commutative). Let's try it here.
- Set the cursor at 3 on the A scale, representing sqrt(3).
- Align the left index with the hairline.
- Move the cursor to 2 on the C scale.
- Read the result under the hairline on the D scale: 3.46.
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| Left index on A:3, cursor on C:2, result at D:3.46 |
Method #3: If you recall from the Basic Multiplication section on this site, the CI scale provides an alternative approach, in which we divide by a reciprocal (which is the same as multiplying by the original number). This provides a great way to avoid index-swapping. Let's give it a whirl, still using commutativity to help us out.
- Set the cursor at the first 3 on the A scale, representing sqrt(3).
- Align 2 on the CI scale with the hairline (indicating division by 1/2).
- Move the cursor to the right index.
- Read the result under the hairline on the D scale: 3.46.
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| Cursor at A:3, then flip |
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| CI:2 to the hairline, cursor to right index, result at D:3.46 |
If you're using a Rietz type rule (which is simplex), then no duplex-flip is required. Here's what the above computation looks like on the Aristo 803:
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| Cursor to A:3, CI:2 to hairline, cursor to right index, result at D:3.46 |
Something to note from these three methods: none of them required an index-swap. That won't always necessarily be the case for all operands, as we'll see in the next exercise.
And, hey, do you need me to remind you yet again that you can click any of these pics to see them full-size, and thus catch the details?
Exercise 1.2: Find the product of 2 and sqrt(30):
Recalling that the A and B scales are bipartite, with the first half of their span for square roots of arguments with an odd number of digits, and latter half for square roots of arguments with an even number of digits, we better see another example. Again, there are three approaches possible (at least).
If you try 2 times sqrt(30) directly as we did in the first approach to Exercise 1.1, above, you'll bump into an index swap at once. The second operand, sqrt(30), is off-scale to the right. If you didn't see this coming, do an index swap and carry on:
- Move the cursor to 2 on the D scale.
- Align the right index with the hairline.
- Move the cursor to 30 on the B scale (3 on the second half).
- Read the result under the hairline on D:10.95.
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| Cursor to D:2, right index to hairline, cursor to B:30, result at D:10.95 |
Let's try commutativity once again with Method #2 to rearrange the multiplicands.
- Move the cursor to 30 on the A scale (the 3 in the second half, of course).
- Align the right index with the hairline.
- Move the cursor to 2 on the C scale.
- Read the result under the hairline at D:10.95.
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| Cursor to A;30, right index to hairline, cursor to C:2, result at D:10.95 |
Notice that both Method #1 and Method #2 here began with the right index. So, possibly some index-swapping was required to get the ball rolling if you weren't fully awake this morning. That won't be the case with Method #3 which uses the ever reliable CI scale for dividing by a reciprocal.
- Set the cursor at the second 3 on the A scale, representing sqrt(30).
- Align 2 on the CI scale with the hairline (indicating division by 1/2).
- Move the cursor to the left index.
- Read the result under the hairline at D:10.95.
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| Cursor to A:30, then duplex-flip |
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| CI:2 to hairline, cursor to left index, result at D:10.95 |
Again, if you're doing this on a Rietz rule no duplex-flip is required.
Wrap-Up: Pondering the six calculations above, it seems to me that the most efficient approach to products involving square roots is always to use commutativity to position the root first, then conclude with the evergreen "divide by a reciprocal" (CI) method. This completely obviates index-swapping false starts, and moreover reminds you to pick the correct half of the A scale at the outset when you're perhaps a little more alert.
Still, for chained operations it's always as well to understand all of the methods, since any one of the "lesser" approaches may reduce the cursor/slide movements in the midst of a complex computation.
One final thought: it occurs to me that all index-swapping and duplex-shifting with square roots would be a thing of the past if AF and BF scales existed...
Next installment: Multiplication by Squares