What is a slide rule, and why was it’s invention such a big deal?
In: 1999
It’s an extremely simple mechanical calculator, based on the fact that log(A) + log(B) = log(A*B), so that a simple addition/ subtraction operation can be used to multiply numbers quickly. A skilled slide rule user could do calculations quickly and accurately enough to support things like manned space flight.
Electronic calculators, let alone a full “scientific calculator” like the ones you get for college courses, weren’t always a thing of course. The invention of the slide rule was big deal because it allowed you to calculate complex equations and large numbers much faster by hand. And by hand was still the only way to do it.
It basically automates the use of log tables to do multiplication and division. I used one in the early 70s to do physics and engineering calculations before I could afford a calculator – which were expensive!
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Multiplying numbers is harder than adding them. You need to do a bunch of smaller multiplications then add them up, and there’s plenty of scope for error.
Maths has a trick (invented by a Scot called John Napier) that lets you turn multiplication into addition. There’s this thing called the logarithm, and if you take the log of two numbers and add them together, it gives you the same answer as multiplying the two numbers and then taking the log of that. In other words log(A)+log(B)=log(A×B). This means you could convert the numbers to logs, add them together (easy) and then convert the answer back.
But, logs are tricky. We can’t easily do them in our heads. So we had to use charts known as log tables. You’d look up log(A) and log(B), add them together, then find the answer in the table to convert back. If this seems like a lot of effort with a lot of scope for error, that’s because it is!
Slide rules simplify this process. Instead of having the numbers spaces out linearly so the distance from 1 to 2 is the same as the distance from 2 to 3, slide rules use a logarithmic scale. This means the distance from 1 to 10 is the same as the distance from 10 to 100. Effectively, the distance along the ruler becomes the log of the number. And slide rules have two of these scales on them which can slide past each other.
So, when you want to multiply A and B, all you do is slide the ruler so the 1 on one ruler is beside the A on the second. That means every number y on the first is now lined up with y×A on the second, because the distances are added together (and the distances are the logs of the numbers). Then you just look along to find B on the first ruler, and the number across from it on the second ruler is just A×B.
This takes complicated, intricate multiplications and turns them into sliding a thing and reading a number!
Edit: thanks, kind stranger 🙂
Edit2: and all you other strangers too. Who’d have thunk so many folk cared about the length of tools used for multiplying. Oh, wait…
To illustrate how big a deal it was for anyone doing complex calculations, I would refer you to this passage about the autobiography of Nevil Shute, who was most famous as a novelist (A Town like Alice, On the Beach, Requiem for a Wren, etc.). He was an Engineer, Nevil Shute Norway.
His autobiography was called [‘Slide Rule’](https://en.wikipedia.org/wiki/Slide_Rule:_Autobiography_of_an_Engineer). He was the chief calculator for Barnes Wallis, doing the stress calculations on R100, a successful airship, scrapped after the R101 (unrelated) disaster.
“The R100 design was the project on which he mentions using a [slide rule ( a Fuller cylindrical model](https://en.wikipedia.org/wiki/Fuller_calculator)), only mentioned once in the book. The stress calculations for each transverse frame required computations by a pair of calculators (people) for two or three months. The simultaneous equation contained up to seven unknown quantities, took about a week to solve, and had to be repeated if the guess on which of eight radial wires were slack was wrong with a different selection of slack wires if one of the wires was not slack. After months of labour filling perhaps fifty foolscap sheets with calculations “the truth stood revealed (and) produced a satisfaction almost amounting to a religious experience”.
Shute later went on to found Airspeed Limited, an aircraft manufacturer, which was famous for the Airspeed Oxford and (after he’d left) the Horsa glider, which put most of the UK & Commonwealth Airborne Force’s heavy equipment on the ground on D-Day, Market Garden and Varsity.
It lets you quickly multiply or divide numbers.
You learned long multiplication and division at school. It takes a bunch of time. Addition and subtraction is much quicker.
There is a mathematical function called “logarithm”, written “log”. It takes a number and gives another number. Exactly how it works isn’t interesting right now. But the interesting part is that adding two logs together is the same as multiplying the original numbers and then taking the log. I.e. (log X) + (log Y) = log (X * Y).
So lets take a stick and put marks along it, like a ruler. “log 1” is zero, so we’ll choose a starting point towards one end of the stick and mark that “1”. Then for every other number X, we calculate P = (log X), and mark the number on the stick at P millimetres from our starting point. Now we can calculate “log X” by measuring the distance along the stick.
OK, just having ONE of those sticks is not very interesting or useful. But if we have TWO such sticks, that makes things interesting.
First, pick two numbers A and B. Find those numbers on your two sticks. Now, line up the sticks so that A on the first stick is next to “1” on the second stick. Like this:
1=============A=============X=====
1=============B======================
On the first stick, read off the number X.
X is at the position (log A) + (log B) millimetres from “1”. We know this because the distance from 1 to A on the first stick is (log A), and the distance from 1 to B on the second stick is (log B). And by simply lining the sticks up, we have added those distances together, so X is a distance of ((log A) + (log B)) from “1”.
But wait, maths tells us that ((log A) + (log B)) is (log (A * B)). So X is a distance of (log (A * B)) from “1”. And because of the way we marked the sticks, we know that the number X is a distance of (log X) from the “1”, too.
So log (A * B) = log X.
So A * B = X.
OK, there was a lot of explanation there, but if you ignore the “why” then actual usage is really simple:
To multiply two numbers, line up one of the numbers on the first stick with the “1” on the second stick. Look for the other number on the second stick, and read off the result on the first stick.
Division is the opposite of multiplication, so:
To divide two numbers A / B, line up A on the first stick with B on the second stick. Look for the “1” on the second stick, and read off the result on the first stick.
A slide rule is just the two sticks described here, attached to each other so they can slide to carry out these calculations.
Actual slide rules may have a lot more features, but the above describes the basics.
Some caveats:
1. “log” only works with positive numbers. But when multiplying or dividing numbers, if one of the number is negative then you can just change it to positive and do the calculations, then change the result to negative. If both numbers are negative you can just change both of them to positive and do the calculations, and you will get the correct result. This is easy enough to do in your head without slowing down.
2. A slide rule has a limited range. What if you need to multiply huge numbers? Well, you remove zeros from the end of the numbers (or shift the decimal point to the left) until you get them small enough to be sensible. Then you do the calculation, then add all the zeroes back on to the result (or shift the decimal point to the right). Similar rules exist for tiny numbers and for division. Again, easy enough to do in your head.
3. Reading numbers off a scale has limited precision. But you can get accurate enough to do some amazing things with just a good slide rule. A lot of engineering can be done with 3 or 4 digits of precision – they have enough safety margins.
It’s a ruler that you can use as if it’s a calculator, once you know how. It’s a zero-technology, battery-less, accurate and fast device that can do complex calculations very simply. It’s basically a few bits of ruler wood, with markings in particular places like on a ruler, where parts slide up and down against each other (hence the name).
But you can literally use it to do the physics calculations that got us to the moon, to the accuracy that was necessary to do that, from something that you can carry in your pocket or your toolbox.
I haven’t seen anyone discussing converting numbers to powers of 10, which is an incredibly useful thing (add needed for using a slide rule).
If you wanted to multiply 12,567 by 200,768, you can convert these to 1.2567 x 10^4 and 2.00768 x 10^5. The power of 10 is how far you have to move the decimal point to make it an integer followed by the fractional part.
Now, you can get a fairly close approximation by multiplying the integers and adding the powers of 10 together: 1.25 times 2 is 2.50; adding the powers of 10 gives you 10^9.
So, 2.50 x 10^9 (or 2,500,000,000) is a pretty good approximation to 2,523,051,456.
On a slide rule, you would do your best to eyeball and line up 1.2567 on one scale and 2.00768 on the other scale, and probably come up with a number much closer than 2.50.
Take two 1 foot rulers, and place them side by side so the the 1inch marks line up. Now slide one of the rulers so that the end lines up with the 3 on the other ruler. You can now see all the additions of 3 by looking at how the numbers line up.
|….1….2….3….4….5….6….7….8….9….10…11…|
|….1….2….3….4….5….
You now have a simple slide rule that you can do simple addition and subtraction with.
Now instead of evenly spaced marks, you use a logarithmic scale you can multiply and divide numbers based on the fact that log(a) + log(b) = log(a+b)
Sorry if the above doesn’t format just right, but you should be able to get the basic idea
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