Several people in the geekout thread asked me to explain how a sliderule works, and I’ve been meaning to write a couple of article about manual computing devices. So I thought I’d do it. There’s a nice slide-rule simulator at Derek’s Virtual Slide Rule Gallery, which is what I used to generate the images in this article.

I know a lot of people think that the idea of learning to use something like a slide rule is insane in an age of computers and calculators, and that this is a silly thing to post about. But I really *love* slide rules, and not *just* because I’m a geek. Slide rules make math *tactile*. Using a slide rule makes you understand how certain kinds of math work; and not just a theoretical understanding, but an understanding on a very concrete, physical level. My dad taught me to use one not because I needed to know (I’m not that old!), but because *he* loved it and thought it was cool; my slide rule is the one that he used in college. He gave it to me when I was in high school. It’s a beautiful K&E log-log duplex decitrig.

There are a couple of things to be said about slide rules up front. They’re beautiful things, and the guy who invented them is an incredible genius. But they’re not a tool for the weak-of-heart. Using a slide rule isn’t like using an electronic calculator. You actually need to do an approximation of the calculation in your head, because the slide rule doesn’t do powers of ten; you need to do that by yourself! Also, in general, the slide rule is used for the “hard stuff”; multiplication and division, logarithms, exponents, and trigonometry. Addition and subtraction you do by yourself, either in your head, or on paper.

The basic idea of the slide rule comes from logarithms, in particular this fundamental identity: x * y = b^{logb(x) + logb(y)}. That is: adding logarithms is equivalent to multiplying numbers. The slide rule places numbers onto a ruler on a *logarithmic* scale; so the distance from “1” to a number “n” on the rule is the logarithm of “n”. That’s the whole fundamental trick to make it work.

Let’s take a look at a slide rule. This is a picture of a Pickett Microline sliderule. That’s a very simple rule, which is easy to see on the computer, but it’s relatively wimpy. It doesn’t have a lot of scales (which is equivalent to a calculator with very few buttons); and it’s really only good for 2 to 2.5 significant digits. (Personally, I’m not a pickett fan; I prefer the big old K&Es, but that’s just because they’re what I’m used to.)

For multiplication and division, we only need two scales: the D scale, which is the top row of the lowest third of the rule; and the C scale, which is the bottom row of the moving slide in the center. C and D are done with the same logarithmic scale. We’ll also use the *cursor*, which is the vertical line on the transparent view slide.

Let’s say we wanted to multiply 22.5 by 3.7. We move the center slide so that “1” on the C scale lines up with 2.25 on the “D” scale below it:

Now – since adding logarithms is multiplying numbers, and the position of a number on the C and D scales are determined by the same logarithm, that means that “3.7” on the “C” scale is in the same position as “2.25*3.7” on the D scale. So what’s on the D scale at 3.7? We slide the cursor over (both to mark the position, and to make it easier to read), and find that it’s at 8.3.

So the answer is 8.3 times 10 to the something. The rule doesn’t tell us what. So we do it approximately in our heads. It’s about 20 times 3 and a half, which is around 80. So the answer is 83. (The exact answer is 83.25, but this rule isn’t big enough for us to see that.)

See? Simple. Now, if we wanted to multiply that by, say, 18, we’d slide the “1” over so that it lined up with the cursor… Except that then, the answer is off the end of the rule. But no problem! There’s *also* a one on the *other* end of the rule. We can slide the C scale so that its *right hand* 1 is over 83 where we’ve left the cursor. Now we slide the cursor down to 1.8 on the C scale:

And you can see it’s sitting at about 1.49. But since we only used two digits, we can only read two digits, so we say 1.5. Now we need to do our powers of ten: it’s about 20 times 80, which is 1600. So it’s 1.5×10^{3}, or about 1500. (Exact result is 1494.)

Division is almost the same thing done backwards: x/y = a^{loga(x) – loga(y)}. So, to divide x by y, we put “y” on the C scale over “x” on the D scale, and slide the cursor over to 1 on C. For example, let’s take π/2. Most rules have a specific mark for π to make that easy. So we slide 2 on C to line up with π on D:

And slide the cursor to one on C:

And our answer is: about 1.57. (The cursor is about half-way between the marks for 1.56 and 1.58; and π is positioned to three significant digits.) We need to do the powers of ten for division to, but that’s easy; we know π/2 is between 1 and 2, so it’s 10^{0}, so the answer is just 1.57.

What’s the real answer? About 1.5708.

See? Isn’t that cool?

[sr]: http://www.antiquark.com/sliderule/sim/