Before computers and calculators, there were slide rules. It is difficult for people today to appreciate just how magic it was to be able to carry a small tool, made of bamboo and plastic, that could perform many of the computations of engineering and science which used to be so tedious in mere seconds, as long as you were happy with its limited precision.

In my own years in engineering school, I did almost all of my homework and examination calculations with a slide rule and, when I graduated and was able to buy one of the first generation of pocket calculators (an HP-45), was amazed by its precision and ease of use, but had to admit that in practical work I could get the job done just about as fast with a slide rule.

The mathematical magic behind the slide rule is logarithms. Logarithms, invented in the 17th century, allow reducing multiplication to addition and division to subtraction. It is very easy to add and subtract by sliding two rulers against one another, so if you use rules whose scales are logarithmic instead of linear, you can multiply and divide without all of the tedious calculation, as long as you’re satisfied with the accuracy you can read off the scale. With even more cleverly defined scales, it is possible to compute the squares of numbers, square roots, cubes and cube roots, trigonometric functions, and logarithms and exponents.

A slide rule does not add or subtract. Humans add and subtract, and with a little practice they get very good at it. (There is a famous scene in the film Apollo 13 where engineers start fiddling with their slide rules to add and subtract numbers. You can identify the balding engineers of my generation as those who chuckle in the theater when this happens.) A slide rule also doesn’t keep track of decimal places. It works only with the significant digits of numbers, and it’s up to you to figure out where the decimal point goes in the result. This sounds tedious, and in the examples that follow it may seem so, but in fact users of slide rules could usually place the decimal by instinct, and if you got it wrong the result was usually so ridiculous you could go back and figure out where you’d goofed.

A slide rule is only as accurate as its manufacture and calibration and your ability to set it and read values. In practice, you get between two and three significant figures from slide rule calculations, and these errors compound as you use intermediate values in subsequent calculations. This may seem crude in an era where effortless 15-digit double precision arithmetic is at everybody’s fingertips, but the fact is that it’s all you need for many calculations in science and engineering. To a large extent (apart from trajectory calculation and mission planning), it got us to the Moon.

### Turnip Truck

You have a truck which can haul a maximum load of 1000 kg (1 tonne). You’re asked to haul a load of 35 sacks of turnips which weigh 28 kg each. Will the truck be able to handle the load?

You whip out your slide rule. You move the cursor to 3.5 on the D scale (recall that the slide rule knows nothing of decimal points; that’s up to you). You align the start (1 mark) of the C scale on the slide with the cursor.

You can now read off anything multiplied by 3.5 by reading from the C to the D scale, so you move the cursor to 2.8 on the C scale.

Looking at the D scale, you read 9.8. From the magnitude of the numbers you multiplied, you can easily set the decimal point to the result of 980 kg; the truck can carry the cargo. (How did I set the decimal point? Well, 35 and 28 are around 30, and 30 times 30 is 900, so the result has to be of that magnitude.)

Now this may not seem very magical, but that’s because you’ve never worked out problems like this by long multiplication and division. Being able to get the answer out in a second or two, even with the minor mental gymnastics of setting the decimal point, was a wonder when folks of my generation first encountered it.

Now let’s move on to some more complicated problems which show off the capabilities of this implement of math instruction.

### Baseball

Let’s start out with a simple physics problem: “How long would the energy of a fastball pitched at 100 miles per hour keep a 100 watt bulb burning if converted to electricity with 100% efficiency?”

The equation for the kinetic energy of a moving body is given by:

The mass, m, of a baseball is around 145 grams. Since we’re asked to compute power in watts, an SI unit, it’s easier to start by converting everything into SI units. The mass of the baseball is, then 0.145 kg; this can be worked out trivially in the head. Now we need to convert miles per hour into our value for v in metres per second. Consulting a table of units tells us that a mile is about 1609 metres and an hour is 3600 seconds, so to get v in metres per second we need to compute:

v = (100 miles/hour) × (1609 metres/mile) × (hour/3600 seconds)

Now recall that we don’t keep track of decimal places when using a slide rule: that’s done mentally or, if it gets messy, with pencil and paper. So, our conversion factor from miles/hour to metres/second is 1609/3600 which, since we only compute to an accuracy of three significant figures, we round off to 161/360. To perform the division, set the cursor to the 1.61 mark on the D scale. Now move the slide to move 3.60 on the C scale under the cursor.

Move the cursor to whichever end of the C scale falls within the D scale. The result of the division, 4.56, can be read off the D scale.

Since by inspection, 161/360 is somewhat less than a half, the decimal point is placed to yield the result, 0.456.

Now we can proceed with the rest of the calculation. First, we need to square the velocity v. Since the velocity was given as 100 miles per hour and we now know the conversion factor, we obtain the velocity as 45.6 metres per second simply by mentally shifting the decimal point two places to the right. To square this value, move the cursor to to 4.56 on the C scale (note that you don’t really need to move the cursor, but simply move the slide to the right so the C and D scales are aligned, since the cursor is already at that point on the D scale).

Now flip over the slide rule and read the square, 2.01 from the B scale. Since it’s obvious that the square of 40 is 1600, we can place the decimal point to give a value of 2010.

Now we multiply by the mass, m, which is 0.145 kg. Move the cursor to 2.01 on the D scale, then move the slide so the 1 on the C scale lines up with the cursor.

Move the cursor to 1.45 on the C scale and read the product, 2.91, from the D scale.

Since we’re multiplying 2010 by a value on the order of 1/10, the decimal point is placed to give a product of 291.

Finally, we need to multiply this value by ½ or, equivalently, divide it by two. We already have the value set up on the D scale, so move the slide to bring 2 on the C scale under the cursor.

Move the cursor to the 1 on the C scale and read off the final result, 1.45.

Since we’re dividing a value around 300 by two, the decimal place is set to give 145. The units of this final result are (kilogram metre²)/second², or joules. A watt is just one joule per second, so the answer to the original question is that if the energy of the baseball pitch were converted entirely into electricity, it would keep the 100 watt bulb burning for 1.45 seconds.

A computation of this quantity with a modern computer and double precision arithmetic gives a result of 1.4489 seconds.

### Cannonball

The heaviest field artillery piece used by Napoleon’s army had a bore diameter of 11.5 centimetres. If it fired spherical iron cannonballs, how much did they weigh?

The formula for the volume of a sphere is:

Given the 11.5 cm diameter of the bore and the need for the cannonball to fit down it without jamming (neither cannons nor cannonballs were made with great precision at the time), let’s assume the cannonballs had a diameter of 11 cm. We need the radius, which we can immediately write down as 5.5 cm without resort to the slide rule. Now we need the cube of this quantity. Move the cursor over 5.5 on the D scale. The cube may then immediately be read on the K scale as 166. Note that the K scale has three decades. The cursor falls within the third decade, indicating the placement of the decimal point.

Now we want to multiply this number by π. Set the cursor to 1.66 on the D scale. We can now immediately read the product of 5.22 from the DF (D Folded) scale, which begins with π. Since the original number was 166, we write the product as 522.

Now set the cursor to 5.22 on the D scale. Move the slide so that 3 on the C scale is under the cursor.

Move the cursor to 1 on the C scale. The quotient of 174 can be read from the D scale, but there’s no reason to bother with it, since we’re now already set up for the multiplication. Move the cursor to 4 on the C scale. The volume may now be read from the D scale as 696. The decimal position is obvious from the magnitude of the original number before the division and multiplication.

Now we know the volume of the cannonball, 696 cm³. We turn to the Junior Woodchucks’ Guidebook to look up the density of iron, which is given as 7.87 g/cm³. To compute the mass of the ball, slide the 1 on the right side of the C scale above the cursor, then move the cursor above 7.87 on the C scale and read the product, 5.47 from the D scale. By inspection of the quantities multiplied, we place the decimal point to give a mass of 5470 grams, or 5.47 kg.

Now, recall that the customary unit of mass, the pound, is around 0.454 kg. Divide the mass we computed, 5.47 by 0.454 (you know how by now, using the C and D scales), and we get around 12 pounds. That’s why this gun was called a “twelve pounder!”

Double precision calculation with a modern computer gives a volume of 696.9 cm³, 5.487 kg or 12.098 pounds.

### Interstellar

Now let’s take our slide rule to the stars—the nearest star: Alpha Centauri, 4.37 light years from the Sun. Let’s assume we have a way to accelerate our probe to 90% of the speed of light. How long will it take, measured on-board, for the probe to arrive at its destination?

At speeds close to that of light, the time dilation of special relativity becomes important. Time on board a ship moving close to the speed of light passes slower by a factor of:

Now, let’s get the answer. To simplify, we’ll assume the time of acceleration to cruise velocity is negligible compared to the length of the cruise and that the probe will not slow down at the destination.

We can use any units we wish when calculating with a slide rule, so here we’ll adopt units where the speed of light is 1; since everything is a fraction of the speed of light, it simplifies things.

First of all, how long will it take, as measured on the Earth, for the probe to arrive at Alpha Centauri? This is easy: the distance is 4.37 light years and the velocity is 0.9 times the speed of light. Set the cursor on the D scale to 4.37 and move the slide to align 9 on the C scale above it.

You can then read out the transit time in Earth co-ordinates as 4.85 years by moving the cursor to the end of the C scale.

Now we need to figure out how much time will have elapsed on board the probe. Since we’re computing in fractions of the speed of light, we first square the velocity, 0.9 c. Align the cursor with 9 on the D scale and read the square, 8.1, from the A scale.

Since the quantity squared is close to 1, we place the decimal point to yield 0.81 as the quotient. We subtract this from 1 by hand, yielding 0.19.

Now we require the square root of this quantity. Set the cursor to 19 on the A scale (using the second decade on the right due to the magnitude of the quantity), and read off 4.35 on the D scale as the square root.

This is the factor of time dilation which, placing the decimal point, is 0.435. We can now multiply the length of the trip as measured on Earth to obtain the time measured on board the ship. Move the cursor to 4.85 on the D scale (transit time as measured on Earth), then move the slide to align 1 on the right side of the C scale with the cursor.

Finally move the cursor to 4.35 on the C scale to read off the ship’s clock time, 2.11 years, from the D scale.

A modern computer calculation yields a transit time as measured from Earth as 4.86 years and time measured by the ship’s clock as 2.12 years.

### Explore!

If you want to further explore this forgotten skill, you’ll find many vintage slide rules for sale on eBay (don’t become a collector; it can ruin your life!). The basics of slide rule computations are explained in this tutorial. The International Slide Rule Museum has a multitude of resources for slide rule fans, including scans of user manuals from a variety of manufacturers. Here is a virtual slide rule which works in your browser.

Here are two videos (with regrettable autofocus hunting) explaining the basics of computation with a slide rule.

This is a somewhat tedious 1943 U.S. government training film on multiplication and division with the slide rule.

A 1944 sequel covers more complicated calculations including proportion, percentages, squares, and square roots.

(Photo credits: all photos are by John Walker, and in the public domain.)

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1. Thatcher

You can add values on a slide rule, John. It’s done in a manner similar to the technique known as “counting on one’s fingers.”

John Walker: A slide rule also doesn’t keep track of decimal places. It works only with the significant digits of numbers, and it’s up to you to figure out where the decimal point goes in the result. This sounds tedious, and in the examples that follow it may seem so, but in fact users of slide rules could usually place the decimal by instinct, and if you got it wrong the result was usually so ridiculous you could go back and figure out where you’d goofed.

If you’re an engineer, you go back and figure out where you goofed. If you’re an economist, you write a book, found a School, and win a prize.

• #1
• June 25, 2016, at 11:34 AM PDT
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2. Inactive

Fascinating as usual, John. I’m saving the tutorial and virtual slide rule, even though the chances I’ll ever use them are vanishingly small.

• #2
• June 25, 2016, at 11:38 AM PDT
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3. Member

Great post John. How about a post on the abacus next week, also amazing.

• #3
• June 25, 2016, at 11:39 AM PDT
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4. Member

My dad bought the most expensive Pickett slide rule he could find when he went to college in the 1950s. It was a true work of art, and really more than he needed for his degree (architecture). But, it was free because he was on GI Bill, and GI Bill paid for it.

In high school we were using cheap plastic slide rules in class. My dad let me take his, and it wowed my science teacher. (He tried to buy it from me, but it was my dad’s, so I said no.) By the time I went to college the calculator had arrived. My freshman year (1973) Michigan still offered a 1-credit slide rule use class, but it was discontinued soon after.

Dad still has the slide rule hung up in his architectural office in the basement of his home. He turned 94 two weeks back.

Seawriter

• #4
• June 25, 2016, at 11:44 AM PDT
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5. Member

When I was in the 6th grade, logarithms and slide rules fascinated me. I would spend hours making my own slide rules with slips of paper. (Don’t tell me you can’t add with a slide rule! You just need to make it for that function)

In engineering work, it was really a feature that you needed to be keeping track of the valid range of the result in your head as you went. I think with calculators and computers, engineers have lost that skill. And anyone working with real measurements rarely has more than 3 or 4 digits of accuracy to work with.

My first calculator was a Bowmar 4 function calculator. When I could afford a more elaborate HP, I gave the Bowmar to our son to play with in his crib in the mornings. He is now an Architect – I am not sure there is a connection, though.

• #5
• June 25, 2016, at 11:44 AM PDT
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6. Inactive

I still have my log-log, duplex-decitrig slide rule from high school in the 1950’s. Also started out with a HP 45 as my first calculator. My first actual computer, except for some play-time on the Eniac at the U of Pennsylvania while a high school physics student, and the same on the Univac at the Franklin Institute, my first, actual computer was an Apple, the first version ever, which was a Steve Wosniac hand made version.

These were pioneer days.

I’m now using a Dell Latitude E5530. What a marvelous 65 years this has been.

• #6
• June 25, 2016, at 11:50 AM PDT
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7. Member

WillowSpring: My first calculator was a Bowmar 4 function calculator. When I could afford a more elaborate HP, I gave the Bowmar to our son to play with in his crib in the mornings. He is now an Architect – I am not sure there is a connection, though.

According to my dad, architects are engineers who cannot handle the math, and go on about how much more important art is than science.

He’s allowed to say that because he is an architect. I am allowed to repeat that, not because I am an engineer, but because I am his son.

Seawriter

• #7
• June 25, 2016, at 11:53 AM PDT
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8. Member

SW:

“According to my dad, architects are engineers who cannot handle the math, and go on about how much more important art is than science.”

Actually, my son started as an Engineer, decided he didn’t like the math and switched, so your dad pretty much has it right.

I, on the other hand, thought I would follow my father, who had a PhD in Physics, but got tired of learning about the unknown and invisible and switched to Engineering where I could see the results of my work.

• #8
• June 25, 2016, at 11:59 AM PDT
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9. Thatcher

From the heart of the Jet Age, baby!

The Breitling Navtimer, circa 1954. See the two outer scales? A circular slide rule.

You can still get one. I still want one.

• #9
• June 25, 2016, at 12:02 PM PDT
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10. Member

When I was in law school, I roomed with a guy studying architecture. He bought the first calculator I’d seen. It could add, subtract, multiply, and divide for \$105.

I learned how to use a slide rule in chemistry class in high school.

• #10
• June 25, 2016, at 12:15 PM PDT
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11. Member

This is such a wonderful article! I found a nice slide rule in a nice leather case a few months ago, and was wondering if I could recall using it. I remember so well making those decimal point mistakes in high school, and laying out very long equations and then going through many slides of the rule to get my answer. Too many to be confident, must repeat!

My parents were not well educated, but my dad knew what a slide rule was (though not how to use one), and he was so proud of me when I started using one for my homework. To him, it meant his son was going to be a professional.

I am going to work through your cannonball equations and ponder Napoleon (who’s skills in these matters launched his career) over a beer tonight.

• #11
• June 25, 2016, at 12:15 PM PDT
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12. Member

Great post, John. I’m a physicist in a family of engineers, so Dad had taught me how to use slide rules, even though I was born in 1972, and we had a digital calculator in the house by 1979.

Dad has given me his two, now—the one from high school and the one from college. The college one is a log-log duplex-decitrig, which is fantastic. I used it a little bit in high school calculus class when my calculator had broken and hadn’t been replaced. A couple of days or so—really just to show off.

• #12
• June 25, 2016, at 12:20 PM PDT
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13. Contributor
John Walker Post author

Tim H.: The college one is a log-log duplex-decitrig, which is fantastic.

That’s the model I used in making the illustrations for this article. It’s a Keuffel & Esser model 4081-3. It bears a copyright of 1947, but also a citation to a patent from the early 1950s, so it’s probably fifties vintage. I bought it on eBay some years ago.

In high school and college I used a five inch Lafayette Electronics slide rule which had the same scales (including log log). It was a lot more convenient to carry around, and I never missed the extra precision from the larger size.

• #13
• June 25, 2016, at 12:31 PM PDT
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14. Contributor
John Walker Post author

WillowSpring: (Don’t tell me you can’t add with a slide rule! You just need to make it for that function.)

Well, you can add and subtract with two linear scale rulers, but you don’t have those scales on the usual slide rule. The only linear scale is the L (and on some slide rules Ln), but you don’t have the two sliding linear scales it would take for addition and subtraction.

• #14
• June 25, 2016, at 12:35 PM PDT
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15. Contributor
John Walker Post author

Percival: The Breitling Navitimer, circa 1954. See the two outer scales? A circular slide rule.

Here’s how the slide rule scales work.

• #15
• June 25, 2016, at 12:46 PM PDT
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16. Inactive

This is a great post.

My High School stopped officially teaching how to use slide rules just before I got there, but our Chemistry teacher thought it was important enough that she took all the abandoned school slide rules and the glorious 6 foot model to hang on the chalkboard and sliced a week out of chemistry class to teach us.

I never ended up using any chemistry, but I’ve always appreciated knowing how to use a slide rule. I’ve still got my cheapie plastic model that I bought for myself!

• #16
• June 25, 2016, at 12:52 PM PDT
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17. Inactive

John, your pictures are of the same model as my slide rule. K+E must have made millions of them; I think over half of the slide rules I ever saw were this one.

A few years ago when I was teaching an evening class as an Associate, I held an after-class slide rule workshop for seniors in civil engineering who had never had the opportunity to get their hands on one. We had a blast, and I had to repeat the workshop a couple of times.

• #17
• June 25, 2016, at 1:31 PM PDT
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18. Inactive

it’s up to you to figure out where the decimal point goes in the result. This sounds tedious, and in the examples that follow it may seem so, but in fact users of slide rules could usually place the decimal by instinct, and if you got it wrong the result was usually so ridiculous you could go back and figure out where you’d goofed.

This is something that the old professors have noted. Engineering students these days are more likely to turn in errors because they don’t notice as well when they have gone off the rails.

• #18
• June 25, 2016, at 1:35 PM PDT
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19. Member

Great & fun post John! I went to college in the 1970’s and switched from a slide rule to an electronic calculator as soon as I could afford it (a scientific model was very expensive at the time). Over the years I’ve lost all of my slide rules except for a circular model – a SAMA & ETANI Model EE-112. I liked it because I could fit it into my shirt pocket.

• #19
• June 25, 2016, at 1:37 PM PDT
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20. Member

Gee, I used my slide rule mostly to figure batting averages and ERAs.

My grandfather gave me one well before I started high school. He wasn’t completely sure how to use it (he had only a 6th grade education, so had picked this up on his own) but what he showed me was close enough to get me started. He had also given me some clues as to what logarithms were for, so it didn’t take long to figure it out.

I forget now which HP calculator I had my eye on when I started grad school, but when I went to buy one I learned about the HP 25c, which was better and cheaper. It wasn’t the first calculator I ever used – not even the first HP scientific calculator – but it was the first one I owned. I didn’t realize I was learning to program with it. A couple years later I bought a TI-something-or-other and sold the HP 25C. I still wish I had never sold it. The TI had more memory and could run bigger programs, but it didn’t have that HP keyboard on which you could work quickly.

• #20
• June 25, 2016, at 1:53 PM PDT
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21. Contributor
John Walker Post author

tigerlily: Great & fun post John! I went to college in the 1970’s and switched from a slide rule to an electronic calculator as soon as I could afford it (a scientific model was very expensive at the time).

Until the HP-35 appeared in 1970, and its scientific calculator competitors from Texas Instruments and others subsequently, there wasn’t a pocket calculator which could perform functions beyond the four basic arithmetic operations of the first calculators. With a slide rule, you could square, cube, extract square and cube roots, compute trigonometric functions (sine, cosine, tangent), logarithms, exponentiation, etc. The HP-35 was a miracle, but when it came on the market, it cost US\$ 395. The 2016 ObamaBucks equivalent of this is O\$ 2446, which was totally out of reach of starving students.

• #21
• June 25, 2016, at 1:54 PM PDT
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22. Inactive

The level of accuracy of slide rules conveniently matched the significant figures in the majority of engineering calaculations. Young engineers now practically have to have it beaten into them to not write down all the digits from their calculator or spreadsheet (I’m a young engineer, so I speak from experience).

I actually got a slide rule off Ebay because I find them elegant (though I lost it when moving to a different cubicle–debating getting another). A closely related topic is nomograms–basically single -purpose mechanical calculators on paper, with lines spaced and scaled to solve some tricky equations. I also find those elegant.

• #22
• June 25, 2016, at 1:59 PM PDT
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23. Thatcher

That cockpit looks a lot different now.

• #23
• June 25, 2016, at 2:03 PM PDT
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24. Contributor
John Walker Post author

The Reticulator: The TI had more memory and could run bigger programs, but it didn’t have that HP keyboard on which you could work quickly.

And they never, ever died. Sitting to the left of my keyboard as I type this is an HP 48GX I bought more than 20 years ago which I use daily. Not only was the keyboard feel superb, the legends on the keys were molded all the way through so they don’t wear off over time. The thing looks brand new.

I also have an HP-16C programmer’s calculator which is now more than 30 years old and still works perfectly. I don’t use it as frequently, because I don’t do as much machine language programming as I did back in the day, but it’s still just fine. I think I’ve changed the batteries three times in thirty years.

• #24
• June 25, 2016, at 2:04 PM PDT
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25. Member

Fun post and great comments; thanks to all!

My Mom’s 1942 Kensington High (Buffalo, NY) Yearbook had this writeup of the doings of the Slide Rule Club. They started off by making their own. For me that would be good, because it would break the whole thing down into small parts for a “Slow-Solid” start.

Now I wonder if anybody here can demonstrate how to use “Napier’s Bones.”

• #25
• June 25, 2016, at 2:06 PM PDT
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26. Thatcher

That guy in the cockpit had one of these (or a reasonable facsimile thereof):

I never learned to use one of those. The ones I’ve worked on look more like this:

Even that one is a little old for me.

• #26
• June 25, 2016, at 2:16 PM PDT
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27. Contributor
John Walker Post author

Tenacious D: A closely related topic is nomograms–basically single -purpose mechanical calculators on paper, with lines spaced and scaled to solve some tricky equations. I also find those elegant.

Here is a talk by my friend Joe Marasco at Stanford’s EE 380 seminar about nomograms and Bayes’ theorem, illustrated by an example in medical diagnosis. (Spoiler warning: I’ll probably do a Saturday Night Science about nomograms some time in the future, and use this video, so if you view it now, please don’t complain when I present it again.)

• #27
• June 25, 2016, at 2:21 PM PDT
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28. Member

I grew up with cheap calculators. Heck, for a while I proudly wore one of those multifunction wristwatch calculators back in junior high. But some years back I taught myself how to use a basic slide rule. I found a site that had a PDF file that you could print out to make your own, and I created one on cardstock. I then used it almost exclusively in a General Chemistry course. I used it for a bookmark in the text, and with a bit of practice I got so that I could get answers faster than the kids that were using calculators.

Years ago, the college math department had big 6 foot long slide rule that they used for demonstration purposes.

Minor brag: I’ll multiply two two-digit numbers in my head for fun. Did you first example on my own.

I’ve found that it can be handy to keep some fractions as decimals in your head, and vice versa. If you need to divide by 14, say, you can see that it’s very close to 100/7, so you can multiply x by 7 and then divide by 100. I’ll use that if I want to, say, calculate the gas mileage for my truck.

As my junior high math teacher drilled into us, the 11th commandment is: “Thou shalt not divide, but multiply by the reciprocal.” Words to live by.

• #28
• June 25, 2016, at 2:53 PM PDT
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29. Member

I remember being thrilled to move from log tables to a slide rule. Though I was pretty impressed by log tables too. So modern. So clever.

My husband is a slide rule prepper. I reckon he has five or six and just bragged that he still has his very first slide rule.

Our son just told me he used steam tables in his engineering degree. I have no idea what he means. They’re not used for cooking though.

• #29
• June 25, 2016, at 2:56 PM PDT
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30. Thatcher

Great post. I was introduced to the slide rule as a high school sophomore in 1969. It’s the only reason I survived chemistry.

I still have it.

• #30
• June 25, 2016, at 3:07 PM PDT
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