Many geometric and trigonometric computations involve the constant π, and occasionally its cousin π/4. Because these constants arise so commonly, just about every slide rule includes a gauge mark for the former, and Picketts also provide one for the latter.
But no tears if your stick doesn't have them. Just remember the approximations 3.14 and 0.785 and you're good to go. Let's try a couple problems using them.
Exercise 1.1: Find an approximation to 7.6π.
Method #1: We'll begin with the standard multiplication technique in which C is used to measure along the D scale, understanding that an index-swap may be needed at the outset to avoid any off-scale disorders later on.
- Set the cursor at 7.6 on the D scale.
- Align the right index with the hairline. (The left index puts things off-scale.)
- Move the cursor to π (3.14) on the C scale.
- Read the result under the hairline on the D scale: 23.9, after adjust for the order of magnitude.
My Pickett 1006-ES includes the required gauge mark. Click the photo if you need to enlarge it.
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| Right index at D:7.6, cursor on C:π, result at D:2.39 |
Method #2: Let's try the ever-handy "dividing by a reciprocal" approach using the CI scale. We'll have to commute the two operands, since I have yet to see a slide rule which places a π gauge mark on the CI scale.
- Set the cursor at π (3.14) on the D scale.
- Align 7.6 on the CI scale with the hairline.
- Move the cursor to the left index.
- Read the result under the hairline on the D scale: 23.9, after adjusting for the order of magnitude.
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| CI:7.6 aligned with D:π, result under index at D:2.39 |
Method #3: Our third tack using the folded scales is the most efficient of all, since it requires no slide movement! Here's the deal.
If you stop to think about it for a moment, the folded DF scale, which is offset by π units with respect to the D scale, already represents a multiplication by π. Just recall Napier's rule about the sum of the logs corresponding to the log of the product. In other words, the DF scale automatically augments things by π logarithmic units.
For example, D:1 aligns with DF:π , D:2 aligns with 6.28 (about 2π) and so forth. So, to solve Exercise 1.1 now becomes a snap, and the slide is never used:
- Align the cursor with 7.6 on the D scale.
- Read the result under the hairline on the DF scale: 23.9, after adjusting for the order of magnitude.
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| Cursor to D:7.6, result at DF:2.39 |
If it's the case you'd really rather prefer the result to land on the D scale (say for a chained operation), just divide DF:2.39 by CF:1, followed by a scale-transfer to read off the result under the index on D. But we're getting ahead of ourselves, since we haven't looked into division yet, not even by something as trivial as unity! So hang on...
Computations involving π/4 pop up fairly commonly. As mentioned, Pickett slide rules thoughtfully include a gauge mark for this, although it's pretty easy just to remember its numerical approximation, 0.785. I've only seen this mark on Picketts, and it's easy to miss if you don't know what to look for:
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| Note the tick mark at 0.785 |
Let's try it out in a problem. We'll use the traditional multiplication method.
Exercise 1.2: Find a decimal approximation of 3π/4.
- Set the cursor at 3 on the D scale.
- Align the right index with the hairline. (We could see the off-scale disorder coming and eschewed the left index).
- Move the cursor to π/4 on the C scale.
- Read the result under the hairline on the D scale: 2.36.
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| Cursor on D:3, right index to hairline, cursor to π/4, result at D:2.36 |
Wrap-Up: For multiplication by π, pretty clearly Method #3 is your best choice, unless an impending operation within a chained computation suggests otherwise. As for π/4, well, that really is a pretty pedestrian number, all things considered, and any of the usual multiplication methods will no doubt perform just fine. I rarely give it any thought until I try to flick the debris off of 0.785 and then realize it's a gauge mark!
And this concludes the collection of techniques for special products. We can now handle multiplication by squares, square roots, cubes, cube roots, π and π/4, and had an introduction to arguments raised to the 2/3 and 3/2 powers as well. All this with some pretty trivial scales available on just about any slide rule!
Coming soon to a theater near you: how to deal with arbitrary exponents using the miraculous log-log scales.
Next installment: Three Methods for Reciprocals
Next installment: Three Methods for Reciprocals