How do users of slide rules deal with overflow outside [1, 10]?

Question

The other day I explained slide rules to a guy who wanted to use them to help his woodworking projects but didn't know how to use one.

I was able to do a good job explaining it, but I cheated. I used the A/B series which I know are there more for historical purposes, rather than the high precision scientific-notation C/D series. So the idea was, woodworking is usually done in quarters or eighths or 16ths or 32nds of an inch, I wanted to show him how you didn't need to move the slide rule once you had found the results of a division, the 1 of B is already pointing at some result on A, that means you can immediately read off B=2, B=4, B=8, looking for where you see that come close to an A line, “oh it's over 4.6, almost 4.7", now look here that times 8 it's smack dab between 37 and 38, well 4×8 is 32 and 37 is 5 more than that so it's a bit over ⅝", call it 4 and ¹¹/₁₆".

Here's the deal, if I didn't cheat and used the C/D scales like a good 1970s physics student, then when I went to multiply by 8 it would overflow, it's 37.5 which is way above 10 and needs to be normalized back down to 3.75 shifting a power of 10 into the exponent.

The only way I see to handle this is to throw in an extra division, so for instance to multiply by 8 I could multiply first by 1.6 which is 8/5, then divide by 2 to get 8/10... Or if that order doesn't work, divide then multiply. The problem there is that there's a lot of human fallibility injected when I shift that 1.6 to 2, I have to mark it with my thumbnail or something and really not twitch because if I skip a little to either side suddenly that's an error injected into the computation...

I am thinking that it must have a more elegant solution than this, especially because the thing that you want to multiply by might be big or irrational and might not suit these sorts of cutesy approximations. Is there a better way to deal with overflow on a slide rule?

Answer 1

Let's say you are doing a multiplication using the C and D scales. Start by writing all your numbers in scientific notation, e.g. $1.6 \times 10^2$ instead of $160$. You get a first estimate of the power of $10$ of the result by adding the exponents of $10$. Each time you use the slide rule for a computation, the slider (the C scale) either extends to the left or the right of the rule. If the slider extends to the left, you add $1$ to the exponent of your initial estimate; if it extends to the right there is no change.

Division is similar, but you subtract $1$ each time the slider extends to the left.

In my experience, most people just took a swag at the size of the final answer and used that to guestimate the exponent of $10$. But a few "highly trained" individuals used the method of left extensions.