Several people in the geekout thread asked me to explain how a sliderule works, and I've been meaning to write a couple of article about manual computing devices. So I thought I'd do it. There's a nice slide-rule simulator at [Derek's Virtual Slide Rule Gallery][sr], which is what I used to generate the images in this article.

I know a lot of people think that the idea of learning to use something like a slide rule is insane in an age of computers and calculators, and that this is a silly thing to post about. But I really *love* slide rules, and not *just* because I'm a geek. Slide rules make math *tactile*. Using a slide rule makes you understand how certain kinds of math work; and not just a theoretical understanding, but an understanding on a very concrete, physical level. My dad taught me to use one not because I needed to know (I'm not that old!), but because *he* loved it and thought it was cool; my slide rule is the one that he used in college. He gave it to me when I was in high school. It's a beautiful K&E log-log duplex decitrig.

There are a couple of things to be said about slide rules up front. They're beautiful things, and the guy who invented them is an incredible genius. But they're not a tool for the weak-of-heart. Using a slide rule isn't like using an electronic calculator. You actually need to do an approximation of the calculation in your head, because the slide rule doesn't do powers of ten; you need to do that by yourself! Also, in general, the slide rule is used for the "hard stuff"; multiplication and division, logarithms, exponents, and trigonometry. Addition and subtraction you do by yourself, either in your head, or on paper.

The basic idea of the slide rule comes from logarithms, in particular this fundamental identity: x * y = b^{logb(x) + logb(y)}. That is: adding logarithms is equivalent to multiplying numbers. The slide rule places numbers onto a ruler on a *logarithmic* scale; so the distance from "1" to a number "n" on the rule is

the logarithm of "n". That's the whole fundamental trick to make it work.

Let's take a look at a slide rule. This is a picture of a Pickett Microline sliderule. That's a very simple rule, which is easy to see on the computer, but it's relatively wimpy. It doesn't have a lot of scales (which is equivalent to a calculator with very few buttons); and it's really only good for 2 to 2.5 significant digits. (Personally, I'm not a pickett fan; I prefer the big old K&Es, but that's just because they're what I'm used to.)

For multiplication and division, we only need two scales: the D scale, which is the top row of the lowest third of the rule; and the C scale, which is the bottom row of the moving slide in the center. C and D are done with the same logarithmic scale. We'll also use the *cursor*, which is the vertical line on the transparent view slide.

So, how do we use this guy to multiply two numbers?

Let's say we wanted to multiply 22.5 by 3.7. We move the center slide so that "1" on the C scale lines up with 2.25 on the "D" scale below it:

Now - since adding logarithms is multiplying numbers, and the position of a number on the C and D scales are determined by the same logarithm, that means that "3.7" on the "C" scale is in the same position as "2.25*3.7" on the D scale. So what's on the D scale at 3.7? We slide the cursor over (both to mark the position, and to make it easier to read), and find that it's at 8.3.

So the answer is 8.3 times 10 to the something. The rule doesn't tell us what. So we do it approximately in our heads. It's about 20 times 3 and a half, which is around 80. So the answer is 83. (The exact answer is 83.25, but this rule isn't big enough for us to see that.)

See? Simple. Now, if we wanted to multiply that by, say, 18, we'd slide the "1" over so that it lined up with the cursor... Except that then, the answer is off the end of the rule. But no problem! There's *also* a one on the *other* end of the rule. We can slide the C scale so that its *right hand* 1 is over 83 where we've left the cursor. Now we slide the cursor down to 1.8 on the C scale:

And you can see it's sitting at about 1.49. But since we only used two digits, we can only read two digits, so we say 1.5. Now we need to do our powers of ten: it's about 20 times 80, which is 1600. So it's 1.5×10^{3}, or about 1500. (Exact result is 1494.)

Division is almost the same thing done backwards: x/y = a^{loga(x) - loga(y)}. So, to divide x by y, we put "y" on the C scale over "x" on the D scale, and slide the cursor over to 1 on C. For example, let's take π/2. Most rules have a specific mark for π to make that easy. So we slide 2 on C to line up with π on D:

And slide the cursor to one on C:

And our answer is: about 1.57. (The cursor is about half-way between the marks for 1.56 and 1.58; and π is positioned to three significant digits.) We need to do the powers of ten for division to, but that's easy; we know π/2 is between 1 and 2, so it's 10^{0}, so the answer is just 1.57.

What's the real answer? About 1.5708.

See? Isn't that cool?

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So, let's make sure I got that right. If I was multipying 2 by 3:

1) Move the 1 line on the C-scale to 2 on the D-scale.

2) Move the cursor to 3 on the C-scale.

3) Read where the cursor is on the D-scale.

If I was dividing 2 by 3:

1) Line up 3 on C and 2 on D.

2) Move the cursor to 1 on C.

3) Read where the cursor is on D.

Hmm. This doesn't seem to work in this case, because it would put the cursor off of the slide rule. Can you only do stuff that gives an answer greater than 1?

Xanthir:

Remember: the two 1s are equivalent. So if the left-hand one is off the left edge in a division, then the result is under the right-hand one.

So if you slide 3 on C so that it sits directly above 2 on D, then Cs left-hand one is off the edge of the rule; but Cs right hand one is sitting on almost 6.7.

Working with a slide rule, you always have to remember that *it doesn't do powers of ten*. On a slide rule, you're working in scientific notation, but doing the 10s exponent by yourself. So 2/3, 20/3, 200/3, 2000/3 are all the same on the slide rule. There *is* no result of division whose first digit is less than 1; using scientific notation, you *can't* get a first digit less than one *unless* your result is exactly zero - and you only get an exact zero if you either start with a zero, or multiply by a zero - and you can't do either of those on a slide rule. (Nor would you *want* to; the slide rule is for things you can't do easily in your head. You don't need a slide rule to multiply by 0!)

Ah, yes, I forgot. If it goes off the slide rule, it just wraps around and increments or decrements the power of 10.

All right, I got it. Thanks!

Yes, exactly! You've got it.

For these reasons, I wonder if slide rules would still be a useful educational tool. Make students estimate orders of magnitude on their own, have them do the "easy stuff" themselves, and use slide rules as a lead-in to logarithms.

Apparently some people still do make slide rules.

I cherish the one unit of college credit I got back in the 70s for my slide rule class my freshman year. I have stashed in my office a classroom demo slide rule that I lug to classes whenever logarithms are the top to be introduced. It's about five feet long, so it's a modest version of the wall-mounted slide rules that used to dominate science and math classrooms in olden times. Those suckers were easily eight feet long.

My father, an engineer, bought the HP35 the year it came out, for about $400 as I recall in 1971 or 72. Until then he used a slide rule and taught me to. It was fascinating for the reasons you describe. What was really fun was going to work with him on Saturdays and seeing the large sliderules (two-feet long) used to get more significant digits -- 4 I think instead of normal 3 you could get from a regular one-foot long.

My favorite though was a circular rule which meant you needn't worry about your calculation "wrapping".

What cannot be appreciated in pictures is the fine craftmanship of a high-quality sliderule. The excellent ones were metal with the markings finely etched. The slide moved easily but was not loose -- you sure didn't want it to slip once you completed a long computation -- and the glass in the cursor was really glass not clear plastic (it also magnified).

Think of it: his company used sliderules to design industrial boilers and heat exchangers used in nuclear submarines and powerplants -- all with sliderules.

I am old enough that I grew up with slide rules and log tables. Mark's calculations actually highlight one of the weaknesses of the slide rule, their inherent inaccuracy! Alright, a good slide rule in the hands of a good operator can deliver fairly accurate results but the emphasis is very much on the double 'good'. As a teenager I had two left hands and ten thumbs so my attempts at getting anything like an accurate answer out of my very elegant Faber-Castell were a total disaster and so I became the fastest user of four figure log tables in the west! I was still slower than my competitors who had mastered the slide rule but I could always beat them hands down on accuracy.

Three years ago I gave a talk to a group of students doing an MA in history on the history of calculation starting with counting on finger, notches in sticks etc and going through to Charles Babbage and the invention of the computer. At some point in the preparation for this talk it occurred to me that my audience although all adults were so young that they almost certainly would never have heard of log tables or slide rules, the greatest Renaissance contribution to the art of calculating. For them logarithms are just the inverse of exponential functions and being humanities student something that was rapidly disappearing into the mist of forgetfulness. For almost four hundred years (2014 is the 400th anniversary of John Napier's first publication) the logarithm served the mathematician, astronomer, physicist and engineer faithfully to be then wiped out of existence in a few sort years by the invention of the pocket calculator.

Yes, that does help. Thank you!

Many of the airplanes that we see flying our skies were created using the slide rule. The planes that Kelly Johnson created were all created using a slide rule. P-38, SR-71, U-2, etc. Most of Boeing's comercial airplanes were originally designed using a slide rule. 707, 727, 737, 747... The calculator is a fairly recent invention...

The problems of having to move the sliding section are solved when you use a circular slide rule. I still use it (an E6B) regularly for flying purposes. Set your anticipated airspeed against the the enroute time and numbers can be seen at a glance - even when its turbulent. no batteries, always shows the last calculation - pure simplicity!!

Slide rules can be useful in explaining concepts. I once used an "addition" slide rule (2 metric rulers) to explain number lines to an elementry student. It seemed to help a lot.

Long after they became functionally obsolete for day-to-day use, my dad got one of these. If you unrolled the scale off the cylinder, it'd be over 5 feet long.

On another page by the same author, he claims that you could probably do multiplications with an error of about 0.04%, so you'd get 4 significant digits much of the time, with something a lot shorter than a conventional slide rule.

I remember those HP calculators when they first came out. $400 might not sound like a lot now, but just for comparison, I was paying (if I recall correctly) $130 a month for a one-bedroom furnished apartment in Pasadena. But fortunately I was a mathematics student so I didn't have any need for such a thing.

Not that I was a lightning calculator, it's just that mathematicians don't do much calculating involving numbers.

I just got another chance to use the K&E <4090-3> that was my Grandpa's.

He used it to:

1. reverse engineer V-2 rockets for the US. Air Corps off trajectory data.

2. build a nuclear plant that supplies northern USA.

I like to think about the bridges (etc.) that I drive across by men & women now gone who didn't worry about "battery life".

Great idea, Mark. I like this topic, and I hope I can keep up as the lessons progress.

Hey Mark, you made it onto the del.icio.us popular list with this post. 52 people have posted it as my writing this.

I handmade my own "sliderule" for work as recently as this past spring. Part of my job involves coordinating curves plotted on log-log graphs (base ten, of course) so that they just barely don't overlap. In order to verify that the margin of safety - expressed in +/- 20% of the x-coordinate - was not violated, I printed off a blank graph at the same scale and trimmed off the margins to get a horizontal strip. If I lined up the tick mark for 1 with the first curve, then the distance from 1 to 1.2 is my margin of safety that I can't violate with my second curve.

Most of the time I don't have to do this (aren't computers grand), but sometimes I don't have the equation for the curve, so I have to compare by hand.

OK, the next thing you'll want to know about is the (medieval) astrolabe. I mean that seriously! If you like slide rules, you'll just love the things you can do with an astrolabe. I teach my history of astronomy students how to use one - I'm not sure how much they get out of it, but I enjoy it...

Error?

"Let's say we wanted to multiply 22.5 by 3.7. We move the center slide so that "1" on the C scale lines up with 2.25 on the "D" scale below it:"

s/22.5/2.25

Nope, that's right. Sliderules work in scientific notation, so if you're multiplying 22.5 by something, you'd use 2.25.

I feel strongly that for many students, math is impossible because they are being forced to try to think abstractly when they are still in concrete operations stage. It's a great pity that math classes are not a lot more hands-on. As you said...

As a private pilot, I found learning the (slide rule-based) Kane Dead Reckoning Computer was very interesting. I wote a post on it here: Kane Dead Reckoning Computer.

I think another thing that people overlook is that computers with software solutions are sometimes wildly inaccurate due to software bugs viruses or whatever.

Sometimes they aren't very far off and so the mis-calculations are not caught ... this can be dangerous when building a rocket, or nuclear reactor.

hello

great post. i will now buy a slide rule!

You should put a link to Using a Slide Rule Part 2: Exponents and Roots at the bottom of this page and put a link to this one at the top of Part 2. Thanks for the refresher.