3.6 Inverse and Folded Scales
One often finds inverse scales on a slide rule, such as CI and DI. The CI scale, for instance, is just the C scale printed in reverse. Since the logarithm of the inverse of a number is equal to the negative of the logarithm of the original number: \[ \log x^{-1} = -1\times \log x = -\log x \] then the CI scale can be used to find the reciprocal of any number on the C scale. This can also be used in multiplication/division problems since dividing by a number is equal to multiplying by its reciprocal. That is, by using the procedures above for multiplying numbers with the rule but by using the appropriate inverse scale, one is actually dividing. This can often help speed up multi-step calculations.
Another aid to multi-step calculations can be the use of folded scales, such as CF and DF. As we saw earlier, sometimes adding or subtracting logarithms leads to an answer that is off the end of the rule. So, rather than moving the slide to the left or right to realign during a complex calculation, one might opt to use the appropriate CF or DF scale; these scales have the identical numerical spacings as the C or D scales and so perform the same function, but they begin and end in the middle of the normal C/D scale range.
The geometric middle of the C or D scale (which goes from 1 to 10) will be equal to the square root of 10 = 3.162. Many slide rule manufacturers realized early on that it might be more convenient to use \(\pi\) = 3.142 as the fold-point rather than \(\sqrt{10}\), thus creating an easy way to multiply by \(\pi\) which is quite often useful, and this became common on the more modern slide rules.
Let’s do a few of examples:
Multiply 2.5 times 6.2 times 9.4. Let’s start by aligning the index on C with 9.4 on D, and moving the cursor to 2.5 on C. To perform our next multiplication we could reset the slide by aligning the left index on C to the cursor. However, our next factor – 6.2 – would be off-scale on C. If, instead, we put the index of CF at the cursor, then the 6.2 is on-scale on CF. Moving the cursor to 6.2 in CF thus gives us our final answer under the cursor on D: 146.
Multiply 1.5 times 3.7 times 6.4. Using the D and CI scales, align 3.7 on CI with 1.5 on D. This is similar to our “division” technique, but here we are dividing by the inverse of 3.7 and so it is actually a multiplication. The answer, still, would be found on the D scale below the index on CI. But, the index on CI is aligned with the index on C. Thus, without even moving the slide, we can multiply by our third number by moving the cursor to 6.4 on the C scale and finding the final answer below on the D scale: 35.5.
Multiply 7 times 9 divided by 5: To illustrate our next technique, think of the problem as solving for \(x\) where \(x=7\times 9/5.\) Now re-write our equation as a ratio, namely \(x/7 = 9/5.\) If we set 5 on the C scale opposite 9 on the D scale, we have set our “ratio”. Without moving the slide, if we look for 7 on C, we should find \(x\) opposite it on D. However, we see that the 7 on C is off scale to the right. But all is not lost. Still without moving the slide, we look up at the CF scale, and we find that the 7 is on scale and opposite, on the DF scale, we find our answer for \(x\): 12.6.
One might also find combinations of inverted and folded scales on some rules, such as CIF or DIF. The Aristo slide rule pictured below contains CI and CIF scales as well as DF/CF scales, for example. This gives the user many options for performing a series of multiplications and divisions. .
In fact, with a full set of these scales – C, D, CI, CF, DF, CIF – on a linear slide rule, it is possible to perform sequential multiplications and divisions without the need to “reset” the slide during the process. Answers at each intermediate step will be available to the user on either the D or DF scale. And note that if one has a circular slide rule, there will be no need for folded scales at all. Nothing goes off scale during a calculation when using a circular rule.
The sequence of numbers on these inverse and folded scales are just the same as those on the standard C and D scales, but start at a different “phase” or end-point, or they are read in an opposite direction. Some manufacturers called slide rules with the CI scale a polyphase or maniphase rule (particularly by K&E and Dietzgen, respectively). Note that the A/B and K scales should be called “harmonics” of the base scale. It’s the CI scale, the CF/DF and CIF scales, and other similar scales and combinations that are truly the same scale with an offset (or, different “phase”) and hence it is these that create a so-called multi-phase slide rule.