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Website last updated 03 August, 2010

Folded Scales

Take a look at the CF and DF scales. They're just like C and D, except they only have one index, 1CF,1DF, in the middle. So if you multiply with CF and DF, and the result is offscale, there's nothing you can do, you can't reset to 10CF - there is no 10CF.

Now try to multiply 2 x 2 with CF and DF. Set 1CF at 2DF, then look for 2CF. Offscale. Same with 3 x 3.

In fact lots of multiplications will be offscale with CF and DF. Likewise, when perfoming divisions the index itself might go offscale. Useless scales huh! Read on..

It turns out there's a useful pattern to the offscale values:

When a multiplication result is offscale on CF and DF, that same result will be available onscale on C and D without resetting.

When a multiplication result is offscale on C and D, that same result will be available onscale on CF and DF without resetting.

That's their purpose. No more resetting the slider when values go offscale. This is especially important when chaining double multiplications together with the CI scale. If you include a CIF scale to complement the CF and DF scales, you can chain any number of multiplications together, jumping up between the C,D,CI group and the CF,DF,CIF group when intermediate results are offscale. This is important because any reduction in number of movements reduces accumulated error and decreases calculation time.

Warning!

Those statements in the instruction box are only true if you use a bit of common sense as well. If you're asked to multiply 9 x 2 and you push 1C all the way to the right end of the rule and set it on 9, well, 2 is definately going to be offscale on C and on CF. You'll be lucky if the slider doesn't fly out and get someone in the eye. The rules apply as long as you choose the minimum motion available. Keep 1C as close to 1D as you can with every choice. If 1C or 10D "cross the center line", all bets are off. In this case you could have chosen to push 1C to 2, looking up 9 on CF, or obviously: 10C to 9 and find 2 on C.

Examples:

( we're going to dispense with the decimal point calculations in these examples. They've been covered extensively in the multiplication tutorial, and nothing is different )

41 x 26
It's hard to estimate whether the answer will be >10. We'll guess an use the 1C index.
set 1C on 2.6 D
notice that 4.1 C is just offscale
read answer on DF at 4.1 CF
answer is 1.066 -> 1066.
There's no need to set the cursor on 4.1 CF on the slide rule shown in the above image, but not all slide rule configurations have CF and DF right on top of each other. If your slide rule has other scales in between CF and DF, you'll need to use the cursor.
3x4x9x3x2 using double multiplication with the CI scale:
3x4: set 4CI over 3D, intermediate result 1.2->12 at 1C on D.
x 9: 9 is offscale on C. Read it on DF at 9CF instead: 1.08 -> 108.
Set Cursor at 9CF to start another double multiplication.
x 3: set 3 CIF at cursor. Intermediate result is offscale on DF at 1CF, read it at 1C: 3.24->324.
x 2: read final result on D at 2C: 6.48 -> 648.
Notice we jump up to the CF side of things when we go offscale down at C, then continue working up on CF until we go offscale, jumping back down to C. You can continue multiplying chains of any length with this method. Remember that using CI to multiply is exactly the same procedure as using C to divide, so you can throw divisions into you chain as well. If you do use divisions, notice that the folded group is upside down compared to the C group - reciprocal method division won't actually look like a reciprocal up on the folded groups.

CF, CF_10 comparison

CF starts at pi, which is 3.14, and CF_10 starts at sqrt(10), which is 3.16. There's less than 1% difference between those numbers, so why bother?

If you use scales folded at sqrt(10), you'll never go offscale

If you use scales folded at pi, you'll go offscale 1% of the time, but you can also use them to multiply/divide by pi with no extra motion

In science and engineering, factors of pi are very common, whereas sqrt(10) appears less often. It's up to personal preference and/or the brand of slide rule you're emulating. The default rules all fold at pi, but you can make your own folded at sqrt(10). If you're using extended scales, such as on the Griffenfly T1, you won't go offscale at pi either.

It probably doesn't need to be stated that it would be incorrect to use a CF (pi), DF_10 pair for the onscale multiplication instructions described above.

Additional Notes

The scale names for CF and CF_10 both appear as CF, but you can check the folding on the autodoc card, or the design screen.

The other folded scales: CFM, CF/M are not folded particularly close to the center of the decade, and aren't as useful for onscale multiplication. They're for converting between log base e and log base 10 depending on the type of LL scales you use.

Other types of folding - at 3.6 for instance, aren't available in the Griffenfly scale set.