I’ve wanted to write about this since the abacus post, but I didn’t want to do two calculating posts too close together. According to my calculations, an acceptable interval has now elapsed, so here we go!
Older readers, you may think this post isn’t for you, but even if using a slide rule is etched into your muscle memory forever, you may have forgotten exactly why the thing works in the first place. I won’t claim to teach anyone how to become a slide-rule virtuoso, but I do think it’s fascinating to know how it works, so I’ll focus on that and keep it pretty simple.
The main idea behind the slide rule is the logarithm, which was introduced in 1614 by John Napier—Jost Bürgi developed a similar concept independently around the same time but didn’t publish until 1620 (Newton and Leibniz were such copycats!). There’s evidence of logarithm-like mathematics as far back as the Babylonians around 2000 BC, but Napier’s work is generally considered the start of modern logarithms. In any case, his work was quickly followed by the idea of the common logarithm and the first log table in 1617, both thanks to Henry Briggs, and then the invention of the slide rule itself by William Oughtred around 1630.
Briefly, logarithms as we use them today are defined by reversing the concept of exponentiation, which deals with powers, or the basic formulation that
where x and y are numbers we know, and finding z is our object. To reverse this, imagine starting with z and knowing the number x you want to take to some power y. Now we’ve reframed it so y is our unknown quantity:
This is the basic concept of the logarithm—taking the starting and ending numbers in an exponential operation and finding the factor that links them.
For most people’s purposes, logarithms are either based on the number 10 or the number e, and called common logs or natural logs respectively. Here’s how a common log works:
As you can see, you can express the common log of a number a as, “What power do we need to take 10 to in order to end up with a?” That means we can also say
or the common log of 10 to some power is just that power itself.
When working with logs, we can also use some properties of exponents. One of those is that multiplying the same number taken to two different powers is the same as adding those powers:
Think of taking 23 x 22. You’re really just taking (2 x 2 x 2) x (2 x 2), so you might as well say 25.
This property, together with the previous definitions of the log, means that using common logs, we can look at the product of two numbers a and b like this:
This means that if you have some way to put numbers on a scale that corresponds to their log value, you can add two numbers and get a value that tells you their product instead!
Enter the slide rule, which has all kinds of scales that make a lot of sense once you realize this is the underlying idea. For instance, here are the C and D scales on a typical slide rule:
This is actually a log scale, which means that the placement of each number represents not its own value on the number line between 1 and 10, but the log of that number on a scale from 0 to 1. Here’s what I mean:
This explains both the odd scaling and the fact that it starts at 1 instead of 0. It’s always nice to understand the reasoning behind details like that.
From the properties we discussed above, we should be able to use these scales to multiply numbers. If log (a x b) = log a + log b, then using a log scale should mean we can add a and b on the scale (so actually log a and log b) and get an answer that represents a x b (actually log (a x b) on the scale).
(If you’ve noticed by now that I’m a nerd, you won’t be surprised that I think this is really cool, but I hope you do too. It’s an amazing idea! There’s a huge shift in perspective from looking at numbers as regularly spaced points on a line, which is a natural mental image, to visually reimagining them based on a different scale. It’s a great way to turn an somewhat abstract idea into something very concrete.)
Let’s try out the slide rule! I’ve illustrated how you would multiply numbers by two, following these basic steps:
1) Move the slide (the inner part) to the right until the “1” on the slide (C) scale lines up with your first number on the bottom (D) scale. This puts you in position to add a number on the C scale to what you have on the D scale.
2) Find your second number on the C scale. Spatially speaking, this is adding that number to the first one.
3) Determine what number that lines up with on the D scale below. This is your answer!
Here’s how it looks when the slide is set to multiply 2 by various other numbers. I’ve marked where each factor (2 through 5) lines up with the bottom scale and how the spatial addition actually lets us multiply the numbers.
Don’t take my word for it! Use this virtual slide rule to try it out yourself and see how the calculations work out. You can do decimals with the more detailed scale as well.
As you can see with the virtual one, an actual slide rule has more scales, including A and B, which allow you a greater range, since they go up to 100. Here I’ve just given an introduction to the concept of the log-based slide rule scale, because I think it’s a cool thing to know and a great example of how mathematics and creative thinking can combine to make life easier.
If you’re interested in learning more about how you can use a slide rule not only for multiplication (and division!) but also for square and cube roots, trigonometry, and exponents, I recommend this thorough guide from the International Slide Rule Museum.
Here’s a common log table for numbers 1-10 like the one Briggs came up with back in the day. You can also read up on the history of logs here or in the Wikipedia entry. Log scales are incredibly useful for looking at non-linear changes and are used to measure loudness (decibels), seismic activity (Richter scale), acidity (pH), and stellar magnitude. You can even get log graph paper!