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

Let’s start with the idea that makes the whole slide rule work. I want you to picture two ordinary rulers, side by side, with numbers marked evenly along them. That even spacing is called a linear scale. Now, suppose you had to add 3 + 2. You put the zero of the top ruler against the ‘3’ on the bottom ruler, as in Figure 4.2. Then you look down from the ‘2’ on the top ruler and read off the ‘5’ on the bottom. That’s addition by alignment. Here’s the clever bit: with just that one alignment, you haven’t only solved 3 + 2. You’ve set up a machine that can add 3 to any number. Look at Figure 4.3. With the zero of the top ruler still on the ‘3’ of the bottom, you can read off 3 + any other number just by looking down from that number on the top scale. One setting, many answers. And you can subtract too. In Figure 4.4, you 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’. So the same alignment trick does addition and subtraction. Now, keep that idea in your head, because we’re going to change the problem. Suppose you had to multiply 4 × 8, but you’d never learnt multiplication. Another way to tackle it 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, you add their indices. That’s the key rule: multiplication becomes addition of exponents. Now, if we put our numbers on a ruler again, but this time on a logarithmic scale instead of a linear one, the length of the ruler is scaled in proportion to the indices. Look at Figure 4.5, which shows the logarithmic scale versus the linear scale. On a logarithmic scale, equal distances represent equal ratios, not equal amounts. That’s exactly what we need, because now we can use our rulers for multiplying and dividing instead of adding and subtracting. And that is much more useful. Let me show you how. To multiply by 2, you align the ‘1’ on the top scale with the ‘2’ on the bottom scale, as in Figure 4.6. Then, by simply reading off the top scale at any point, you can find that number multiplied by 2 on the bottom scale. So if you look at ‘3’ on the top, you read ‘6’ on the bottom. One alignment, and the whole scale is multiplied by 2. Now suppose you want to multiply by 3 instead. You don’t need a different scale. You just align the ‘3’ on the bottom scale with the ‘1’ on the top scale, as in Figure 4.7. You’ve now set the scales in the ratio 3:1. So ‘1’ is aligned with ‘3’, ‘2’ is aligned with ‘6’, ‘3’ is aligned with ‘9’, and so on. The same two scales do all numbers — you just change the alignment. Note that it works with the positions of the two scales reversed. You can have your two scales like Figure 4.8, or like Figure 4.9. The orientation doesn’t matter; the principle is the same. And just as the linear scale could subtract as well as add, this logarithmic scale can divide as well as multiply. When set up like this, you can divide 6 on the top scale by 2 on the bottom scale, and read off the answer ‘3’. That’s Figure 4.10. Division is just the reverse alignment. Finally, here’s the step that gives us the actual navigation computer. The Navigation Computer turns our straight slide rule into a circular one, as shown in Figure 4.11. Instead of two straight rulers sliding past each other, you have two circular scales rotating relative to each other. The same alignment logic applies, but now it’s wrapped around a circle, which makes it compact and easy to use in the cockpit. So the whole foundation is this: linear scales add and subtract by alignment; logarithmic scales multiply and divide by alignment, because multiplication of numbers is addition of their indices. The navigation computer is just that logarithmic slide rule, bent into a circle.