The Navigation Computer - Slide Rule Face — Page 71, Lesson 79

I want to walk you through the slide rule face of the navigation computer. This is the part that lets you multiply and divide quickly without doing the arithmetic longhand — and it all starts from the idea of a simple ruler. Let’s begin with addition. Suppose you had to add 3 + 2. You take a straight linear ruler — the kind where the spacing between numbers is even, like a normal ruler. You put the zero of the top ruler against the ‘3’ on the bottom ruler. Now look down from the ‘2’ on the top ruler, and you read off the ‘5’ on the bottom ruler. That’s your answer. Now here’s the clever part: with just that one alignment — zero on top against 3 on the bottom — you can also read off the answer to 3 plus any other number. You don’t need to re-align for each sum. If you want 3 + 4, you look down from the ‘4’ on the top ruler and read the bottom. If you want 3 + 7, you look down from the ‘7’. One alignment gives you all those sums at once. You can also subtract. Align ‘4’ on the bottom ruler with ‘1’ on the top ruler. Then look back to the zero on the top ruler, and you read off the answer ‘3’ on the bottom. That’s 4 minus 1. Now, keeping that idea in mind, let’s deal with a different problem. Suppose you had to multiply 4 × 8, but you had never learnt to do simple multiplication. Another way to tackle this is to treat it as 2² × 2³. That’s the same as saying 2 × 2 × 2 × 2 × 2, which is 2⁵. So 2² × 2³ = 2⁵. In other words, to multiply two numbers expressed as powers of the same base — here the base is 2 — you add their indices. The index is the little superscript number, the exponent. 2 plus 3 gives 5. Now, if we put our numbers on a ruler again, but this time on a logarithmic scale instead of a linear scale, the length of the rulers is scaled in proportion to their indices. On a logarithmic scale, the distance from 1 to 2 represents the same proportional increase as the distance from 2 to 4, or from 4 to 8 — because each step multiplies by the same factor. That’s different from a linear scale where the distance from 1 to 2 is the same as from 2 to 3. Because the logarithmic scale spaces numbers according to their indices, we can now use our rulers for multiplying and dividing instead of just adding and subtracting. That is much more useful for navigation calculations. Let me show you how multiplication works. To multiply by 2, we align the ‘1’ on the top scale with the ‘2’ on the bottom scale. Then, by simply reading off the top scale at any point, we can find that number multiplied by 2 on the bottom scale. For example, if you look at ‘3’ on the top scale, you read ‘6’ on the bottom — that’s 3 × 2. If you look at ‘5’ on the top, you read ‘10’ on the bottom. This works for any number. Suppose we now want to multiply by 3. All we do is align the ‘3’ on the bottom scale with the ‘1’ on the top scale. We have now set the scales in the ratio 3:1. We do not need a different scale for each number — the same two scales do all numbers. Now ‘1’ is aligned with ‘3’, ‘2’ is aligned with ‘6’, ‘3’ is aligned with ‘9’, and so on. Note that it works with the positions of the two scales reversed. You can have the top scale as the one you read from, or the bottom — the principle is the same. We can use it for dividing as well as multiplying, in the same way that we could use the linear scale for subtracting as well as adding. When set up like this, we can divide 6 on the top scale by 2 on the bottom scale, and read off the answer ‘3’. Now, the navigation computer takes our straight slide rule and turns it into a circular one. Instead of two straight rulers sliding past each other, we have two circular scales that rotate relative to each other. That’s the slide rule face you’ll actually use in flight.