The Slide Rule | Hacker News Reader

The Slide Rule

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162 pointsrefrigerator132 comments

I've noticed that my daughter, despite being an engineering student, isn't nearly as comfortable with the properties of logarithms as I am. Of course, she studied them in Calculus classes, but they were just abstract functions to her since she always had computers or calculators to perform calculations. For me, they were a part of the standard education for technical or science related fields. Even in high school we were expected to use slide rules or tables of logarithms to perform calculations.

I bought my first slide rule around 1964 (with money I made delivering the Detroit Free Press newspaper every morning before school). It was a bright yellow Pickett like those available on EBay[1]. It was possible to do multiplication, division, roots, trig, exponentials, etc. to just less than 3 significant digits of accuracy. If this wasn't enough accuracy, I had to recast the computation as addition or subtraction of logs from log tables, which could with some interpolation yield around 6 significant digits of accuracy.

I just recently pulled out my old slide rule to show my accountant how they worked.

At MIT I saw circular slide rules, and some of my fellow students used them (more accuracy on the outer scales, less on the inner most scales). The most interesting slide rules I've seen are cylindrical. The scales wrap around the cylinder in a helix and are much longer giving an extra digit or two of accuracy (but requiring more mental work on the user to figure out which reading of the cursor to use because the cursor was parallel to the axis of the cylinder). See [2]

By the time I was finished using my slide rule, I'd started working with computers. Even today, I have a fancy overpriced TI calculator that just sits in a drawer. I'd much rather fire up a Python REPL to do most calculations.

[1] https://www.ebay.com/itm/VINTAGE-1959-PICKETT-DUAL-BASE-SLID...

[2] https://ocw.mit.edu/courses/edgerton-center/ec-050-recreate-...

I bring out the slide rule pretty much these days to give history lessons to young folks. Last time I showed how it worked to a pair of GenZ kids they looked at me with a healthy bit of skepticism. Like somehow this was some sort of fake contraption. I felt a bit like my late father explaining what it was like to crank a car to start it.

I know what you mean. I borrowed a car from a fraternity brother to pick up a date once. Before departing he explained that if it didn't start I might have to use the crank! It was a Citroën if I remember correctly. I'm sure it really impressed my date.

Extra battery (the one you take inside where it's warm over night) and starter cables in the footwell :)

I do think the first encounter people have with logarithms ought to be doing reverse lookups in exponentiation tables, by ocular linear search. It's not convenient at all, but it should give a good grasp of the concept.

Yes, everyone should learn to use a slide rule in high school, and should spend at least a couple months regularly using it for calculations in math/science class.

As you say, they not only build strong intuition about logarithms, but also help teach about working precision, etc.

Handheld electronic calculators should be banned as having almost no pedagogical value. In general assigned problems should be conceptually more challenging but should have simpler numerical values. Anything that needs more calculation than that should be done on a computer with a real keyboard and a real programming language.

In primary/middle school, counting boards (like those used in ancient civilizations or medieval Europe) should be used much more than they currently are. Experimenting with these tools is great preparation for thinking about how computers store and manipulate data.

I grew up long after slide rules went out of fashion, but I really like them. And not only for their signaling value; I genuinely think they make working with specifically ratios easier.

For example if you have a recipe calling for 150 g of butter, but you want to use all of your 250 g, you can just align 150 and 250 on the slide rule, and then read off the correct amounts of the other ingredients, without ever touching the slide rule again. No multiplication performed.

I understand people don't use slide rules for more things, but I'm baffled they aren't considered standard kitchen equipment, in one shape or another.

I'm not great with mine, but I really love using them. I took some of my engineering finals with it and finished well and quickly.

They just make the calculation more "real" and I feel like I'm actually working out the problem instead of just punching numbers into a thing that gives numbers back.

That wouldn't work in the US, because we measure a lot of things by volume, don't use the same units for everything (small volumes are measured in teaspoons or tablespoons, larger volumes in cups), and tend to use lots of fractions in cooking. So a recipe wouldn't call for 150 g of butter, it would call for 2/3 of a cup. I suppose you could have multiple scales on a kitchen slide rule though.

For American-style recipes you could use something like this: https://www.reddit.com/r/woodworking/comments/5y1r4h/kitchen...

Wouldn't that all still be the same ratio though? If you set the slide rule to convert 250 to 150, shouldn't it e.g. also convert 2 cups to 1.2 cups?

Measuring cups aren't graduated in 10ths. They're split into 8 liquid ounces. 0.2 cups = 1.6 oz.

(My wife, who is from Finland, thinks the measuring system is insane. At least now I'm good at estimating conversion from F to C.)

Your wife is correct. Imperial measurements are absolutely ludicrous.

The US uses customary measures. The UK used Imperial.

TIL

+- 20 % error, insignificant for the accuracies used with either system.

Volume is just base 2 instead of base 10. It’s probably the most sane of the imperial measurements.

I assume you mean the cups that come in various containers of a particular size, because the ones I know come with a scale.

My liquid measuring cups are glass and have a customary scale graduated in liquid ounces, not 10ths of a cup. They also have metric measures on the other side.

Yep. And it would convert 2 tablespoons to 1.2 tablespoons too. There is no reason it wouldnt work for American recipes.

Most American measuring cups don't use decimals, unfortunately. You get things like ¼, ½, and ⅓ cup measures. Of course, still doable, just takes some extra mental work.

And if you can do that much mental work, you probably don't need the slide rule. I think a slide rule graduated with "cooking fractions" (i. e. the sizes of measuring cups and spoons that people actually have) would be a good idea.

https://www.reddit.com/r/woodworking/comments/5y1r4h/kitchen...

Well I use a cooks measure for that allows you to quickly measure different things by volume flour rice etc.

Keep in mind that slide-rules typically have more than one scale. So there could be a scale for volume, a scale for weight- even a scale to estimate cooking times.

Now I want one of these. (Especially the "estimating cooking times" parts - I'm weirdly good at mental arithmetic so I don't actually need this for volumes and weights.)

>>For example if you have a recipe calling for 150 g of butter, but you want to use all of your 250 g,

What a strange way of cooking. I'm not saying it's bad, just that typically people cook for a result (I want to make X servings) not a reason (I want to eliminate all of the butter in my house).

Think of it this way; the last time you cooked this dish there was way too much and youd prefer not to do that again. This would help.with that problem.

I guess it also makes more sense for baking than cooking.

But that's not what this scenario presents. OP was specifically talking about not wanting to leave ingredients around. He's solving a different problem.

All ingredients move together in ratios (got that, totally understand) so I understand HOW this work. What I was commenting on was WHY someone would cook this way. Again, I'm not saying it's wrong, it's just a problem I've never encountered. If I only used 150G of butter and had 100G left, then I have 100G of butter for whatever else I'm cooking.

I'm sure scenarios can be presented for why this is a problem that needs solving (storage space, product expiring, etc) it's never been a way I personally did things.

If you have some ingredients you want to get rid of, make a bigger recipe and have leftovers. If you don't, don't.

I've done this a lot of times.

Basically, the reason is, I want to make as much as possible of something from the ingredients I have. Just find the constraining ingredient, calculate the ratio, and you're set.

But there's so often other restraining conditions. This works better for some things (cookies) than others (a loaf of bread) because of portioning or cooking dishes or other factors.

Again, clearly this is some kind of a thing some people do, I just don't tend to cook this way apparently.

Would you understand it better the other way around? A recipe calling for 250 g of butter, but alas, you only have 150 g left.

Not really? If I have a dinner party involving 6 guests, and my recipe calls for (<ingredients per portion>*6) then scaling down that ratio may give me 4.5 portions.

I don't cook things without purpose/specific goals. I'm cooking to make enough for the people I'm expecting.

That's not to say it doesn't make sense, just not how I've tended to see things cooked.

OK. What if you have a dinner party for 6 guests, but your recipe is written to give 8 portions?

What I'd do in this case (and this is true for most everyone I know) depends on what "portion" means:

- If it's something that can be reasonably divided among guests (such as a soup or something similar), then I eyeball the portions so they're all approximately even. Optionally, if there's an ample amount, you can simply let people decide how much they want through self-service.

- If it's something that's demarcated by physical objects (e.g. dinner rolls, cupcakes, etc), then you have a few other options: Leftovers to save for later, split them among people who want extra, or give them away to whomever wants them.

I don't know about the OP you're replying to, but I rarely divide a recipe up based on expected portions since portion size is highly variable. Should I have extra left over, then I deal with that accordingly. If I'm cooking to get rid of an ingredient, I don't particularly care if I miss the mark by a few portions provided I have enough in the first place for the objective.

(The other problem is that portioning in this question seems to me to assume that all guests are equally hungry.)

That's solving a different problem. Again: I understand how to scale a recipe (which is what you're describing).

Let's do this as a math problem.

How I tend to cook: Xbutter + Ysugar + ZFlour = 20 Cookies.

How OP is cooking: 2 butter + Ysugar +ZFlour = X

We're setting different known quantities (him butter, me portions) and then solving for the rest. They're fundamentally different approaches in cooking philosophy. And again, neither is "wrong" I just found OP's style to be novel enough to comment on

Around our house, we definitely cook with the expectation of having leftovers. While we do plan ahead some, a lot of our meals involve looking in the fridge and trying to figure out something to make with whatever raw ingredients happen to be in there, and hopefully scaling whatever spices/sauces/etc to match. It's only two of us, and we'll often cook 8-10 servings of something to have lunches for the next couple of days.

I do this: recipe calls for 220g of butter, packet I have is 250g; just make bigger batch of cookies.

I just round & then convert in my head though. (Eg 30/220 ~ 15% = a tenth + half a tenth).

I definitely cook this way sometimes, although not with butter. But say I have a recipe that calls for 1 kg of chicken and I have 1.25 kg. I'm going to use all that chicken and scale everything else up by a factor of 1.25, rather than being stuck with that extra 250 g of chicken.

Of course, this only works if you don't mind having too much food. I like leftovers (I get to eat without having to cook!) so it works for me.

This is how baker's percentages work-- you typically start with some quantity of flour like 50 lbs and all the other stuff is available in whatever quantity you need. The problem is getting the ratios right.

No I understand recipe conversion (worked in restaurants for 10+ years) so I understand everything working on ratios. I just rarely in my professional or personal life cooked by saying I have X amount of an ingredient and want to use all of it. I've always started saying "I need X portions of a completed product, and therefor need to have <insert amount of ingredient>).

Replying to you and the GP:

Throwing a spanner into the works, just for fun:

In India (and probably some other countries), cooks often don't measure out ingredients for dishes [1]. They just wing it / use their experience [2] (called andaaz in Hindi/Urdu). And the results are often or mostly good (except with bad cooks, of course).

[1] I guess that approach works for all countries, except individual cases where people blindly follow formulas to the letter.

[2] Of course, they are not flying blind, they use rough (or precise enough) estimates, that are based, again, on their experience, and on just seeing what amounts of what stuff (whether it is pinches, grams, ounces, kilos, pounds, or whatever), makes for a good dish.

Oh sure, but that's a WHOLE other thing. Last night I made potato soup for dinner. I peeled about 2.5 pounds of potatoes, and that was the only thing I specifically had in mind for "will this be enough portions of soup for the people I'm trying to feed" when I started, no measurements or ratios. Everything else was just me leaning on all my experience to make it taste the way I wanted.

That's a whole different style of cooking, and most people don't get there until well AFTER they learned how to cook something with a recipe (or someone telling them how many handfuls of beans to add to the pot).

Good points, and I agree. I meant roughly the same, when I said "use their experience" / "andaaz", although did not make it explicit.

Not all ingredients scale linearly. You would also need a scale exponent.

Slide rules usually have quadratic and cubic scales.

They do by weight.

"A movable pointer called a 'cursor' was developed to make it easier to read numbers off more precisely. (This is likely the origin of the computer cursor!)" Etymonline corroborates this, https://www.etymonline.com/word/cursor#etymonline_v_494. So its cursor was the precursor to our cursor.

I concur, sir.

I got familiar with a slide rule when I studied for my pilot's license, through the "E6B flight computer"[1]. This is essentially a slide rule for specialised calculations (knots-km/h, gallons-litres, density altitude, etc). The E6B is still a mandatory piece of equipment for student pilots but generally regarded as outdated, even by instructors.

As an early millennial I'd never ran into this 17th century marvel called slide rule. When I asked the instructor how it works (rather than how to use it) he answered along the lines of "don't ask, it just does".

[1] https://en.wikipedia.org/wiki/E6B

I got largely the same answer during my PPL. For my check ride the DPE just wanted to make sure I knew how to use it for one simple calculation (estimating fuel remaining after a diversion) then we just used the figures in ForeFlight for the rest of the ride.

Is this the device referred to as the “circular prayer wheel”?

I think the E6B contradicts the article's assertion that the last slide rule in the US was made in 1976; when I was taking flying lessons a few years ago, I was required to buy a new E6B, and I doubt that was from some massive overstock in the 70's.

True. ASA still makes them but I suppose the only reason they're still made is that it's been considered standard classroom equipment for the past 70 years, combined with the well known universal dislike for change in the aviation world. I don't know anyone who kept using the E6B after getting licensed.

The transition was nearly instantaneous. We all used slide rules in high school chemistry and physics. Two years before I entered freshman engineering, “engineering computation” was taught with slide rules. The next year, you could optionally take the traditional course or the new version based around calculators. My freshman year, the calculator-based course was required. As a senior, I bought the most high-end Post slide rule at the campus bookstore at the clearance sale for $3.00.

>The transition was nearly instantaneous.

The first pocket scientific calculator was the HP-35 in 1972 (costing about $2,500 in today's dollars). Within 3 years, at least for most university engineering programs in the US, slide rules were completely gone. [That's incredibly fast.]

In my freshman physics class (1972-73) that price was a brief ethical issue. The professor said that for a midterm we could use one page of written notes and a slide rule. One student asked if he could use a calculator instead of a slide rule. "Are you going to buy calculators for everyone else in the class, too?" the professor asked. He felt it would be unfair to give such a big advantage to those few who could afford an HP-35 at that time.

I’m now a bit curious when the switch flipped at the school I attended. There was no question when I started in 1975 that you needed a calculator. I suspect that those in the class ahead did as well. But that must have been very new.

That IBM ad at the end is priceless on so many levels. (What did that massive computer do? How many female engineers do you count... etc,)

Thanks for the article. My grandfather was an accountant who preferred the slide rule to a calculator. I grew up with slide rules around me, and never could understand the logic.

Another tool lost in the past generation is the Abacus (here is a great story with Feynman[1]) and I would love more info as to how those compared in logic or use, if anyone can shed insight.

[1]: https://www.ee.ryerson.ca/~elf/abacus/feynman.html

I have some of the same experiences. Every older engineer would say "When I was in school we used slide rules" but no one would ever explain how they worked. This article was great for finally clearing that up for me.

This surprises me. I went to school after electronic calculators replaced slide rules (and books of log and trig tables), but we still learned how logarithms worked and that they were the principle behind slide rules.

I graduated from college in 2016. We definitely covered logarithms but nobody really tied it back to slide rules. It definitely is a shame because understanding how problems were solved before can provide a lot of insight.

We used log and trig tables in the 90s (UK) at a time when programmable graphing calculators were widely available.

We learned about logarithms as well in the 2000s, but they never mentioned slide rules. Just lots of tables and charts to get a feel for it.

You can find videos of abacus competitions on youtube, and it's mind blowing how fast they go!

> My grandfather was an accountant who preferred the slide rule to a calculator.

This is hard for me to understand. Slide rules are inherently imprecise, in the same way floating point calculations are imprecise but much moreso. I'd expect an accountant to prefer integer (fixed point) math for the same reason banks do now.

> Slide rules are inherently imprecise

And not particularly good for addition.

Depends on the scales used. With linear scales, they do addition. With logarithmic ones, multiplication.

Sure, in principle. Have you ever seen an actual slide rule with linear scales?

Mine had them. I don't think that I'd prefer it for accounting work, though.

> Mine had them.

I doubt that very much. Do you have a photograph?

I used to collect slide rules. I have never seen one with a linear scale, and I can't find any evidence on the web that any such slide rule was ever manufactured. So if you can provide evidence that you possessed such a slide rule, that would be of significant historical interest.

Isn't the "L" scale on a slide rule linear? See the photo at the top of the Wikipedia page [1] for example. (I may be confused here, since you know slide rules better than me.)

[1] https://en.wikipedia.org/wiki/Slide_rule

Yes, but I'm not sure you could easily use them for addition. They exist to lookup the value of various logarithms. Adding's pretty fast anyway.

Sure, but to do addition you need two linear scales, one on the fixed part and one on the slider. The L scale is for computing logarithms, not for doing additions.

If the L scale is on the slider, and you’re not too much concerned accuracy (the procedure does 5 ‘align scales’ operations before the final ‘read the result’) one linear scale is sufficient, if you use the hairline on the cursor as a kind of memory:

”Example: calculate 0.23 + 0.45

- "Reset" the rule so that all the scales are lined up.

- Move the cursor to 0.23 on the L scale.

- Move the leftmost 0 on the L scale to the hairline.

- Move the cursor to 0.45 on the L scale.

- Reset the rule again so that all the scales are lined up.

- The cursor should now be at 0.68 on the L scale, which is the sum of 0.23 + 0.45.”

(from http://www.antiquark.com/2005/01/slide-rule-tricks.html)

Another possibility:

Using a sledge hammer, break the slide rule into N pieces, taking care to insure that N is greater than the sum you wish to compute. Now to compute the sum of A and B, count out A pieces into a paper bag, then count out B pieces into the same paper bag. Now empty the bag onto a clean workspace and count the total number of pieces to produce the result.

https://www.mrao.cam.ac.uk/~steve/astrophysics/webpages/baro...

I checked, and sure enough you're correct. I could have sworn that I'd had two linear scales, but … I was completely wrong. Been decades since I used it, and the memories obviously faded.

No worries, happens to everyone. But I appreciate you confirming that.

In some sense. A way to teach how a slide rule works is by using two rulers to create a slide rule that does addition. Most pupils will understand why that works. The jump from there to really understanding why logarithmic scales work is a lot harder for most, but I’m sure that intermediate step helps some pupils.

(Google gave me http://demonstrations.wolfram.com/AdditionSlideRule/, but not an example of using normal rulers to make such a slide rule)

> In some sense.

Oh, come on. I'm not talking about a home-brew slide rule or a virtual slide rule simulation. Obviously it is possible to make those. I'm talking about an actual manufactured slide rule sold as a product and used in a working environment to perform addition.

Most slide rules have a linear scale on them, but it is intended to be used along with a logarithmic scale for working with exponential/logarithm functions.

In general slide rules aren’t used for addition because 3-digit addition is very fast to compute mentally or using pen and paper.

Yes, they are imprecise. In general, you're limited to just three, maybe four digits regardless of the actual magnitude of the numbers and you have to keep the powers of ten in your head. Honestly, though, once you've used it enough, it isn't that hard to do. Besides, in most calculations, you don't need precision out to the 12th digit. After all, skyscrapers, bridges, and ocean liners have been built using slide rules for a long time and they, mostly, have worked just fine.

Imprecision is fine for engineering and can be compensated for. But this is not true for managing money, which is why floating point has historically not been used in accounting applications.

Yeah, bean counters do like to know their amounts out to the penny. That would require a pretty big slide rule.

> How many female engineers do you count

Or non-white ones.

The _Hidden Figures_ reminder there that women and people of color were job titled "computers" for a long time in the English language. The term for female/PoC programmers/mathematicians/engineers became the name of the machine built to replace them. It's a fact that feels increasingly strange with distance/hindsight, or at least it should to folks with a heart.

(Just finished Mary Robinette Kowal's excellent "Lady Astronaut" books, and they are a fascinating retro-history of a timeline that fought to get to Mars faster than ours, and an interesting part of that success was getting more Computers in space, but by that it was meant women and people of color that could use a slide rule and do important math quickly in a crisis.)

you are conflating history with sexism/racism, though they overlap. "Computer" was essentially a "technician" who performed calculations, as today you'd have a technician to install parts in your machines or do other mechanistic tasks in an operation. It's not so unusual for a word that refers to a human job eventually becomes the name for the machine that replaces that job: dishwasher, messenger, agent. "computer" happens to be the job that is so easy to mechanise cheaply that the human job was made quickly obsolete once the opportunity arose.

The term "Computer" long predated mechanical computers, by centuries. https://en.wikipedia.org/wiki/Human_computer The earliest "computer" jobs tended to exclude women due to sexism.

This photo shows an example of a human computer team that is mostly men (but may have a higher proportion of women than other technical teams in the larger organization) https://en.wikipedia.org/wiki/Human_computer#/media/File:NAS...

You're more likely to find women and PoC in "computer roles" than in scientist/engineer roles because they tended to be shut out from higher-level roles they may have been qualified for, leading to an oversupply of their demographic at the lower levels. And those overqualified women/PoC have more notable stories than the non-overqualified women and men in those roles.

Appreciate the extra technical details. The conflation is still a useful shorthand in terms of the discussion of the ad in question: it's wording specifically says "replace engineers" with this mechanical computer. You are right, it doesn't need to state "replace [humans] computers with [mechanical] computers" because that is already implied in the name choice.

Whatever the ratio of "over-qualified" human computers were at the time, it was still a non-zero number, and it is useful to remember those "notable stories". Even the "non-over-qualified" members not "notable" enough for memory weren't machines, and you can't claim they had no impact outside of "pure mechanistic tasks"; certainly there was a lot of under-appreciated QA/QC work and inter-personal labor that got shifted around over the years. Any loss of a workforce to automation isn't just "mere" obsoletion, because people aren't machines, and it is good to remember that (conflation, overlaps, or not). Especially for a word like "computer" as opposed to "dishwasher" where we practically never have a reason to "human compute" today, to pull out slide rules and step through some math. (There are still human dishwashers in major restaurants; we still have dishes in our homes that we manually wash in addition to machine washing.)

> What did that massive computer do?

I believe it's an IBM 604 Electronic Calculating Punch. This box was programmed with a plugboard and did BCD addition, subtraction, multiplication and division. It was built from 1100 vacuum tubes.

It wasn't a computer in the modern sense. It read data from a card (the box in front is the card reader), ran through up to 60 steps of calculation, and then punched the result onto the same card, at the rate of 100 cards per minute.

This system was introduced in 1948 and rented for $645 a month (about $5500 in current dollars).

More info: http://www.columbia.edu/cu/computinghistory/604.html

Actually back then the "girls" where the computers imagine typing pool that did maths calculations Los Alamos had a large number to do nuke calculations.

Growing up at a time B.C. (before calculators), we learned to use the slide rule in high school trig. I also used it during my freshman year in college. My dad bought me a Pickett trig slide rule that I still have in a drawer at home. I've taken it out occasionally to play with and I am always impressed by how simple and yet powerful it is. Very elegant. Very dated.

There is a fairly good writeup on various slide rule types and scales, on http://www.quadibloc.com/math/slrint.htm (the rest of this guy's site is worth exploring on its own too, there seems to be a huge dump of various bits of knowledge on it).

A nice reference is Clifford Stoll's (author of A Cuckoo's Egg) article in Scientific American 2006:

http://www.uvm.edu/pdodds/files/papers/others/2006/stoll2006...

And a way to make your own sliderule:

https://static.scientificamerican.com/sciam/assets/media/pdf...

The article has the same IBM ad and says it's from 1953.

People on eBay are selling these Soviet pocket-watch style rotary slide rules:

https://www.youtube.com/watch?v=Kuzdjy3HpWg

I think they were made into the 80s.

Another precursor to logarithms was prosthaphaeresis[1], based on trigonometric functions. Of course, like the quarter-square method, this required lookup tables, not a nifty device. But even with logarithms, log tables were pretty commonly used as well.

[1] https://en.wikipedia.org/wiki/Prosthaphaeresis

Angle measure is just the bivector (‘imaginary’) part of the complex logarithm.

Recently someone posted about a paper-based calculating tool where you had two or three scales and used a ruler to combine the numbers. There was even a Python package for producing them.

What was the name?

Possibly this?

https://en.wikipedia.org/wiki/Nomogram

A Python package for it:

http://pynomo.org/wiki/index.php/Main_Page

This example is one of the coolest things I've seen

https://upload.wikimedia.org/wikipedia/en/a/a5/Risk_Based_Sa...

I had some fun drawing lines and seeing how each factor influenced the chain of isopleths.

Historically I believe they were just "computers" but that is just about impossible to search for these days. "Slide charts" is what I have the most luck with. I've inadvertently picked up a couple (usually tucked into old engineering books I've purchased)so looked into them a while ago.

This page has a great collection of them: http://www.mathsinstruments.me.uk/page66.html

That search term also pulls up a ton on ebay.

I have my grandfather's slide rule. I don't often handle it because it has a patina on it from the thousands of calculations he must have carried out on it. It has several rulings on it and a sliding cursor.

Studying it, I realized that, once set, it shows the multiplication of the whole continuum. In other words, a given setting of the rule to some n shows nm for all m. (Actually all mnEk for all m and k.)

Some slide rules are circular or tubular.

The intro about Kepler is inaccurate, so I distrusted the claim about Newton too. It seems he did use a kind of slide rule: http://www.oughtred.org/history.shtml

> In 1675 Sir Isaac Newton solves cubic equations using three parallel logarithmic scales and makes the first suggestion toward the use of the cursor.

> In 1677, two years after Newton invents the cursor, Henry Coggeshall perfects the timber and carpenter's rule. Newton's cursor fails to catch on at the time.

Thanks for raising this — what part of the intro was inaccurate? This was the source we used for the Kepler info: https://www.jstor.org/stable/27838992?seq=1#page_scan_tab_co...

> Kepler first formulated his hypothesis: planets had elliptical orbits with two foci, and then set out to prove it mathematically.

He didn't start with that hypothesis, he came to it after multiple tries at fitting other models like an equant (an offset circle with angular speed controlled from the opposite offset, which is actually a good first-order approximation to the Kepler-law motion at small eccentricity).

> Only after repeating his procedures 70 times did he offset his computational errors. With a little less patience, he wouldn’t have proved his theory, and elliptical orbits would have eluded us for longer.

I skimmed the paper, and it seems to be about Kepler fitting an equant instead of an ellipse. Apparently he used an iterative method, stopping after 70 iterations? And his numerical errors seem to be responsible for it not converging faster. That makes more sense to me than my misunderstanding that you were saying he ran through the whole calculation from the start 70 times. I jumped to a conclusion about that part, and I'm sorry.

Julian Barbour's The Discovery of Dynamics explains what Kepler did in detail (it's not the whole topic of the book, but a big chunk of it). A great book -- I didn't really appreciate Kepler before reading it. Both brilliant and very human.

> The last slide rule manufactured in the US was produced on July 11, 1976

Actually, ThinkGeek made a production run a couple years back (though I guess I don't know that they were US-made). Doesn't seem to still be on their website, but I have one so it definitely existed. https://www.amazon.com/ThinkGeek-Slide-Rule/dp/B003M5B84C still shows a listing.

I have a couple, too. They were not very good though.

I'd be interested if someone made a new good slide rule. And not just slide rules--there are at least a couple of other older computing devices I'd like a good quality working replica or reproduction of.

1. A Curta calculator. https://en.wikipedia.org/wiki/Curta

2. The Antikythera mechanism. https://en.wikipedia.org/wiki/Antikythera_mechanism

I'm sure you've seen this already, but if not:

https://www.youtube.com/watch?v=j9uRckJLqLk

A 3:1 scale replica of the Curta, 3D printed.

> 1. A Curta calculator. https://en.wikipedia.org/wiki/Curta

The article notes that Marcus Wu has published a printable 3D model of a curta (at 3:1 size due to printer precision issues, though there are comments about people attempting 2:1 builds). You can also find actual curtas on ebay, though they're not cheap ($500~$1500 depending on condition and the like).

You can find all sorts of slide rules on eBay (e.g. K&E). The really good ones can get pricey but they're readily available.

That looks to be a clone of a low-end Pickett student slide rule down to the design of the box. I doubt if they were US-made.

I recall buying one after that in the UK though when I went to work I switch to calculators.

We did still use log tables at college in the late 70's early 80's though the course I was on still had belt drive's in the syllabus - yes the dawn of the industrial revolution type belt drives that ran factories in the 18th century

A practical implementation of a slide rule can be found on watches such as the Breitling Navitimer. I like to use it for quick currency conversions when traveling.

And most are helpfully marked for unit conversions as well! I have mine set for restaurant tipping by default (North America).

I've got a slide rule to thank for my precocious early math studies. I was always interested in math as a young kid, and my parents got me a Post slide rule for Christmas 1970, when I was in 5th grade/10 years old. I didn't know what the S and T scales on the back side of the main slide were for, so went looking up info in the local library. Turns out they were sine/tangent scales, which led to reading about trigonometry, which led to me realizing I needed to know algebra first. Over the next couple years, those library trips led to teaching myself algebra, geometry, trigonometry, and calculus, thanks mostly to TutorText books. For a nerdy little kid in the early 70s, that was heaven.

Reading about the usefulness of logs reminded me of another spatial math tool: The Triangle of Power!

https://www.youtube.com/watch?v=sULa9Lc4pck

This one was seriously awesome! Thanks. Most times when people say Math is hard they are actually talking about this. All the different ways to memorize stuff which doesn't help them get intuitive in Math.

Some of the last people using them for work may have been graphic artists, who called them "proportion wheels". I don't know how many of the folks sizing photos knew that they were using circular slide rules.

At one time there were a lot of industry specific tools to calculate various things (usually made out of cardboard) that were effectively slide rules or slide rules combined with lookup tables.

> The last slide rule manufactured in the US was produced on July 11, 1976

This is bit surprising to me. Does it imply that slide rule production had shifted to other countries, or did it really fall out of fashion so very rapidly? For reference, HP-35 calculator was released only in 1972 (for launch price of $395), and TIs competing SR-50 ("slide rule calculator") in 1974 (at $170). I would expect slide rules to be significantly cheaper and fairly entrenched (especially in colleges etc), so being killed in only few years is remarkable if true.

I'm guessing that's a bit of hyperbole, since new sliderules are still available for purchase today. Obviously manufacture of sliderules dropped precipitously, but there's still a niche market for them.

"manufactured in the US"

I wouldn't be surprised if the quote is accurate.

Missed that. Yes, that makes it plausible as Post and K&E had both ceased production by then and, presumably, most of the cheaper plastic ones were either made in Japan or in places like the Soviet Union for domestic use.

Although that date is all over the Internet, I don't believe it and some detailed timelines for individual companies suggest a few years later. [1] [ADDED: As the sibling notes, I'm sure at least cheap slide rules were available perhaps even today. But that date is plausible for "in the US."]

They were killed off pretty quickly though. When I started college in fall on 1975 I got a TI scientific calculator which was somewhere around $100-$200. (It was required.) Prices were in sharp decline. A five function TI calculator was about the same price the year before. A year or two later an older HP model (which were higher-end calculators) was also about the same amount.

Calculators were fairly expensive at first but remember that slide rules weren't useful for adding and for a lot of calculations anyway. [They're not precise enough for everything.] So once engineers, in particular, switched to calculators, there wasn't much of a market for slide rules.

[1] http://www.sliderulemuseum.com/SR_Dates.htm

1976 was about the time that Commodore made scientific calculators cheap (under $100). Around that time, I remember a Faber-Castell slide rule (a typical engineering-grade device with fine engravings and a cursor that was actually perpendicular to the rule) costing about $30 - and that still left you with three-and-a-bit significant figures, an adding machine for sums and the need to use trig tables. That's pretty much when the student market went away, and anyone who was using alide rules and calculators to make money had little trouble justifying an HP or a TI.

First, the positive note: This is an awesome, accessible article.

Now the error: In the paragraph beginning "By converting numbers into their logarithm", log(4)=2 should be log2(4)=2 - it's been a while since I was in mathematics academically, but isn't it a necessity to show the base of the log if it's not 10?

In general, log(x) does not refer to one base (although 10 is common). In this case, they specify that log(x) is referring to base 2 in the paragraph above, but yeah it could definitely be more spelled out.

It's a convention, not a law. If the base is known (or irrelevant) in context (as it is explicitly laid out for several paragraphs), no need to show the base in every expresion.

The article says: "In the world of 2s, log(x) tells you how many times you have to multiply 2 by itself to get x."

I think it's convention, but it would have been good to explain this a bit more. The previous sentence seems to say "from now on we mean base 2" ("In the world of 2s, log(x) tells you how many times you have to multiply 2 by itself to get x.")

Which base gets to be special depends. Log on its own often means base e; sometimes base 10; and sometimes base 2 (and sometimes 'base whatever, it's not important')

It's an interesting debate in notation. Not all log notations show the base, as they mostly assume you know which base you are in (even/especially in the slide rule era, to my understanding, where the base was always assumed by the slide rule in use; which stories posit was fun when mathematicians would default to natural log slide rules and engineers to base-10 log).

This article would be better if showed the base in its notation, but for the most part the bases are determinant in context of each paragraph, and that's roughly par for the slide rule course.

The base is flexible. A slide rule is fundamentally just a set of adding sticks where you perform addition and subtraction in "log space". As long as you convert back out of the log scale it doesn't matter what the base was.

My primary use for logarithms today is verifying how many bits an intermediate calculation is going to need on a calculator without base-2 logs. The base conversion algorithm doesn't care what intermediate base you use as long as they match: bits ~= log2(x) == log10(x) / log10(2) == ln(x) / ln(2).

This reminds of my uncle who had them in his university and later at work (studying in the 70s in the communist Poland). I remember him mentioning that when first calculators came, skilled slide rule users were much much fasters, than anyone using a calculator.

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