slide rules

Suppose you want to evaluate the multiplication problem $C = A\cdot B$. But you forgot your times tables, so you are stuck using your expedition watch or something. In fact, you can even make your own out of a couple of slips of paper, if you really need to.

First bit of background: One of the rules you probably learned and I definitely forgot was the rule of logs: $$ C = A\cdot B \equiv \log C = \log(A\cdot B) = \log A + \log B$$ This ends up being useful for slide rules, because you can easily add distances together by putting two things next to eachother! So that is exactly what we do. We add the logarithms by putting things next to eachother.

The trick ends up being how we actually label the distances. We pick one convenient unit for the first interval, and that represents going from the value of 1 to the value 2. The key is then using that same interval, but having the value go from 2 to 4. And we continue this way until we have 1, 2, 4, 8, 16, 32, 64, 128 written down the split.

Suppose we call our two halves the left and right scales. Then here’s the setup for actually doing a multiplication: Align the value 1 on the right scale with the value A. Then scan down to the value of B on the right scale, and read off the number we aligned with on the left scale. That is nothing other than C!

If you’re wondering why it is not $\log C$, great question! It’s because every time read a fixed distance, we are looking at the logarithmic scale, but when we read the value, we are taking log or exponential. It sounds weird, but it becomes pretty clear with a bit of practice.

Now, the next problem is: What if we want to multiply by 6, or something else that we haven’t really labeled on our scale yet? The answer is that you end up estimating. One decent estimation once you have all the powers of two is just a linear interpolation. This is basically equivalent to

You can also use this to divide. If you want to evaluate $ Z = X / Y $, just set X (on the fixed scale) next to Y (on the moving scale), and then read off Z (on the fixed scale) by finding the number adjacent to 1 (on the moving scale).