## Monday, December 4, 2006

### Slide Rule Lesson #1

This post, dear readers, is being inflicted upon you at the request of one of my, er, less conventional readers. He actually asked me to explain a bit about how slide rules work, and how you use them.

So I’ll start with the basics. The “scale” you see in the photo (click on it to enlarge it) goes by many different names, depending on exactly where you find it. On the slide rule that I scanned to make this picture, it’s called a “C” scale; that name is just a convention and doesn’t actually mean anything. Virtually all slide rules made after about 1900 used “C” to mean this line. One early device that used this line was called “Napier’s bones"; they were engraved on ivory sticks (the “bones"), and invented by the famous mathemetician Napier. Later they were found engraved on sticks of wood invented by a fellow namd Gunter; these were called “Gunter’s rules” and were in use for hundreds of years.

Whatever instrument you find this line on, you read it from left to right, with the values ranging from 1 to 10 (even though the right-hand side just says “1"; this is another common, though not universal, convention). You’ll note that ticks are not evenly spaced, as they would be on a ruler that was measuring a distance. Instead, the position of the number is proportional to the logarithm (base 10) of that number. The center of the scale isn’t 5, as you would expect on an ordinary ruler — it’s about 3.2. And the logarithm of 3.2 is about .5, which is why it’s halfway across the scale.

Confused yet? Let’s try an example, just to show you how a Gunter’s rule was used. Suppose you wanted to multiply 2.2 times 3.3. You would start by taking a pair of dividers (like a compass, but with two points instead of a point and a pencil or pen). The first step is to lay the divider along the rule from 1 to 2.2 (see the red “divider” on the left side of the photo). Now for step two: take the divider without changing its setting and put the left-hand point onto 3.3 (see the red “divider” on the right side of the photo). The answer — 7.26 — is read a the right-hand point.

What magic allowed this to happen? Well, it’s the magic of logarithms. You can multiply any two numbers together by adding their logarithms and then taking the antilogarithm of the result. You could do this with the giant tables of logarithms found in old reference books, but that’s quite tedious and only makes sense if you need your answer to a quite high precision. But if you only need two or three decimal places of accuracy, the Gunter’s rule could give you the answer quite conveniently. With a little practice, you can multiply two numbers on a Gunter’s rule in just a few seconds, and very reliably. It’s a machine that automates the computing and addition of logarithms, and taking the antilogarithm to get the answer you want.

The same general principle can be used to divide. Using the same photo, we can show how to divide 7.26 by 2.2. The first step is exactly the same as above, setting the dividers for 2.2. The second step is to set the right-hand point of the divider on 7.26 — and then the answer, 3.3, is under the left-hand point.

Modern slide rules perform multiplication and division using exactly the same principles, but use a number of ingenious mechanical aids to make it faster and more accurate than a Gunter’s rule. More about these improvements in a separate post…

#### 1 comment:

1. In the old blog, Simon said:
er, less conventional here,Keep 'em coming. Fascinating stuff!Thanks,Simon