In Chapter 2 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss semilog plots, where the vertical axis is marked using a logarithmic scale. In this case, a constant distance along the vertical axis corresponds to a constant multiple in the numerical value. In other words, the distance between 1 and 2 is the same as the distance between 2 and 4, which is the same as the distance between 4 and 8, and so on. Looking at a semilog plot helps the reader get a better understanding of how logarithms and exponentials work. Yet, what would be a really useful learning tool is not something readers just look at, but something that they can hold in their hands, something they can manipulate, something they can touch.
Enter the slide rule. Sixty years ago, when electronic calculators did not yet exist, the slide rule is how scientists and engineers performed calculations. I didn’t use a slide rule in school. I’m from the first generation that had access to electronic calculators. They were expensive but not prohibitively so, and we all used them. But my dad used a slide rule. He gave me his, mainly as an artifact of a bygone era. I rarely use it but I have kept it in honor of him. It was made by the Keuffel & Esser Company in New York. It is a fairly fancy one and has a variety of different scales.
First, let’s look at the C and D scales. These are marked logarithmically, just like semilog paper. In fact, if you wanted to draw you own semilog graph paper, you could take out my dad’s slide rule, hold it vertical, and mark off the tick marks on your plot axis. On dad’s slide rule, C and D are both marked logarithmically, but they can move relative to each other. Suppose you wanted to prove that the distance between 1 and 2 is the same as the distance between 2 and 4. You could slide the C scale so that its 1 lined up with the 2 on the fixed D scale. If you do this, then the 2 on the C scale really does line up with the 4 on the D scale, and the 4 on the C scale matches the 8 on the D scale. The value on the D scale is always twice the value on the C scale. When you think about it, you have just invented a way to multiply any number by 2.
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| A slide rule showing how to multiply by 2. |
This trick of doing multiplication isn’t just for multiplying by 2. Suppose you wanted to multiply 1.7 by 3.3. You could line the 1 on the C scale up with 1.7 on the D scale, and then look at what value on the D scale corresponds to 3.3 on the C scale. The slide rule has a handy little ruled glass window called the cursor that you can use to read the D scale accurately (if the cursor lands between two tick marks, don’t be afraid to estimate an extra significant figure based on where it is between ticks). I get 5.60. Use you calculator and you get 5.61. The slide rule is not exact (my answer was off by 0.2%) but you can get an excellent approximation using it. If my eyes weren’t so old, or if I had a more powerful set of reading glasses, I might have gotten a answer that was even closer. I bet with practice you young folks with good eyes and steady hands could routinely get 0.1% accuracy.
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| A slide rule showing how to multiply 1.7 by 3.3. |
If you can do multiplication, then you can do its inverse: division. To calculate 8.2/4.5, move the cursor to 8.2 on the D scale, then slide the C scale until 4.5 aligns with the cursor. Then read the value on the D scale that aligns with 1 on the C scale. I get 1.828. My calculator says 1.822. When using the slide rule, you need to estimate your result to get the decimal place correct. How do you know the answer is 1.828 and not 18.28 or 0.1828? Well, the answer should be nearly 8/4 = 2, so 1.828 must be correct. Some would claim that the extra step of requiring such an order-of-magnitude estimate is a disadvantage of the slide rule. I say that even when using an electronic calculator you should make such estimates. It’s too easy to slip a decimal point somewhere in the calculation, and you always want to have a rough idea of what result you expect to avoid embarrassing mistakes. Think before you calculate!
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| A slide rule showing how to divide 8.2 by 4.5. |
Suppose you have a number like 5.87 and you want to know its reciprocal. You could, of course, just calculate 1/5.87. But like most scientific calculators that have a special reciprocal key, dad’s slide rule has a special CI scale that performs the calculation quickly. The CI scale is merely the mirror image of the C scale; it is designed logarithmically, but from right to left rather than from left to right. Put the cursor at 5.87 on the CI scale, and then read the value of the C scale (no sliding required). I read 1.698. I estimate that 1/5 is about 0.2, so the result must really be 0.1698. My electronic calculator says 0.1704.
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| A slide rule showing how to calculate the reciprocal of 5.89. |
One property of logarithms is that log(x2) = 2 log(x). To calculate squares quickly use the A scale (on my dad’s slide rule the A scale is on the flip side), which is like the C or D scales except that two decades are ruled over A whereas just one is over D. If you want 15.92, put 1.59 on the D scale and read 2.53 on the A scale (again, no sliding). You know that 162 is 256, so the answer is 253. My calculator says 252.81. Not bad.
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| A slide rule showing how to calculate the square of 15.9. |
If you can do squares, you can do square roots. To calculate the square root of 3261, place the cursor at 3.261 on the A scale. There is some ambiguity here because the A scale has two decades so you don’t know which decade to use. For reasons I don’t really understand yet, use the rightmost decade in this case. Then use the cursor to read off 5.72 on the C scale. You know that the square root of 3600 is 60, so the answer is 57.2. My calculator says 57.105.
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| A slide rule showing how to calculate the square root of 3261. |
There are additional scales to calculate other quantities. The L scale is ruled linearly and can be used with the C scale to compute logarithms to base 10. Other scales can be used for trig functions or powers.
How to use a slide rule.
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