I love slide rules. The amount of functionality stuffed into such a simple device is beautiful. Also, turning multiplication/division into addition/subtraction with logarithmic scales felt like a genius level hack when I first learned about it, like the fast inverse square.
On a mildly related note, does anyone know if it's possible to construct a logarithmic scale from simple tools? A demonstration I've always wanted to try is building a slide rule "from scratch"
EDIT: by "simple tools" I mean no computers. Pen and paper, compass and straight edge, that sort of thing. I would assume in real life it was made via trial and error on a very large scale and shrunk down optically for a screen print, but I wonder if there's a "precise" way
Yes. You can calculate logarithms by hand using pen and paper. Then, you can make divisions using the compass and a straight edge until you have a finely divided ruler and then mark logarithms on that ruler based on what you calculated.
After all, the slide rule was invented hundreds of years before the computer.
If you want a base-10 scale, I think you only need to calculate the logarithms of 1/5 and 1/2 by hand, and the rest can be done by scaling those ratios (not to scale):
Compass and straight edges compute what are called the constructible numbers [1], which have 0, 1, and anything obtainable via addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers.
These will not allow you to get to logarithms. You can't avoid having to calculate them numerically; from there you may also create approximations to construct them but you might as well just use a ruler at that point.
You can compute logarithms by hand but they're notoriously tedious even by 19th century standards (and they had a great deal more tolerance for that sort of thing than we do today, for obvious reasons), which is why you could buy books full of them, and that's generally what people used.
OTOH the first log table was computed by taking successive square roots of 10. Then you use the binary expansion of each number on your scale, right? So compass-and-straightedge doesn't seem crazy, since square roots are easier that way than numerically by hand.
The original question seemed to be asking about how to make a real chart, that is, with real paper, pens, compass, straightedge, etc. In the real world, taking successive roots is going to bleed accuracy pretty quickly, and when you feed that inaccuracy back in to the e^x function, it's going to bite pretty hard. I don't think it's practical to get the requisite level of accuracy this way.
You'd be multiplying the square roots, not exponentiating.
Yeah, I don't know how practical a construction would be overall for someone who seriously went at it; I was mainly moved to answer because this objection that constructions can't express transcendental numbers just doesn't seem relevant -- digitally you don't keep infinite precision either.
Square roots and multiplying are two of the simplest geometric constructions (geometric mean and taking a proportion), compass and straightedge operations can be accurate, you can construct it enlarged and then scale it down, and finally the precision you need to aim for at the end is bounded. A potentially fun project idea. Think of it as a kind of retrocomputing: how might Euclid or Archimedes have designed a slide rule?
I could be wrong, I haven't had enough coffee yet.
Say you constructed a pantograph where the ratio of the "inner" point to the "outer" point was ... um ... log(5)? for a decimal scale? ( I think you could make a binary scale but I can't think what the ratio would be log2(1)? )
Anyway, you'd start with the pantograph's base and outer point at the ends of your ruler, mark off the inner point, and then repeat on each sub interval, and then again recursively until you ran out of room.
Maybe they focus the "old guy" rays I give off, but so do I. I have a small collection of slide rules, all normal sized linear types, which I still am fascinated with. They're all made between the 1940s and 1960s and aren't terribly rare but the care and precision that went into their manufacture is amazing given that they weren't terribly expensive back in the day. I especially like the one's made of bamboo. Not as much precision nor speed as a calculator or computer but always remember that we got to the moon on them.
I also have a small collection. One is one I found in my grandfather's things. It was a small cheap one, but interesting to me was that it was from an era when computers just starting to supplant people (he worked on a room-sized computer who's job was to count carpool inventory for the army.) I also have one I got from an army-surplus store in a flea market that has various aerial photography calculations tools on the back.
Less precision, more speed than a digital calculator, in my experience (at least in high school physics class in the early 90s). Makes sense because the slide rule takes one precise movement per number but the digital calculator takes one movement per digit.
> always remember that we got to the moon on them.
Well, there were also 5 computers (one analog) onboard the spacecraft, multiple powerful IBM System/360 mainframes computing trajectories on the ground, a Honeywell 1800 assembling code, a whole pile of Univac computers (1230, 494, 495) processing data, and RCA 110A computers for mission control (including one inside the launch platform under the rocket).
True, but my point was that the engineer's who designed the rockets and capsules and towers and just about everything else, including those computers, needed by the space program used their slide rules on a daily basis.
Sure. One undeniable advantage they have is when it comes to scaling multiple numbers by the same factor (for example in the kitchen.) You just set it to the scaling factor and then don't touch it again. Reading off a scaled number is as easy as it gets.
There are other softer benefits too, such as making the quantities in the computation somehow more... visceral.
Don't pilots have them in the cockpit? Easier to calculate remaining fuel on a slide rule than on a calculator, because the analog nature gives a plausibility check.
Just for the sheer heck of it (and because my father got me some polar coordinate graph paper) I spent part of a summer in the early 70s making a circular hexadecimal slide rule.
Back in the day people used printed log tables. Attach your protractor to the graph paper appropriately centered and mark off 360 * log(x) for your base of choice for a lot of values of x.
> [T]urning multiplication/division into addition/subtraction with logarithmic scales felt like a genius level hack when I first learned about it, like the fast inverse square.
Indeed the whole notion of logarithms was developed and introduced by Napier to simplify calculations!
Long ago, I did this. Pick some distance X which is from 1 to 2 to 4 to 8 to 16... to 1024... 1/3 of that distance is almost exactly 10, there will be some approximation involved.
If you pick some given resolution limit, it could all be done by hand, but would take quite a while.
As a follow up, I wanted to know what clever mechanism was used to generate the divisions on slide rulers... I figured someone, somewhere and a linkage that used the fact that the derivative of ln(x) is 1/x... nope... they used a big cam.
It wouldn't really be practical to use paper, compass, and straight edge to construct a slide rule. There are a great marks and they must be quite accurate. The are not spaced evenly; they are spaced logarithmically.
To build a cardboard slide rule get a table of logarithms accurate to around 3 digits and layout a "ruler" with marks for the values from 1.0, 1.1, 1.2, ... 10.0. These marks however are placed on the "ruler" at the location corresponding to the log of the values: 1.0 is placed at the start of the ruler since log(1.0) == 0 and 10.0 is placed at the end of the ruler since log(10.0) == 1. In between the over values are placed where their log says they should go, so for example log(5.0) == 0.699 so the 5.0 should appear on the ruler 69.9 percent of the way from the 1.0 mark to the 10.0 mark.
Two of strips of cardboard labeled in this fashion will give you a very basic slide rule. Placing them next to each other it is very easy to position the slides so that you are "adding" the logs of two numbers. This is how multiplication is done using a slide rule.
Do you mean the Thacher's listing? If so that one's about an 18m equivalent. It's in really nice shape with the original box. I've got the same one, but not as pretty as that one is.
lol, so was I. a slide rule enthusiast myself I own several linear and circular slide rules. that said, I have come across an 8foot x 18inches linear slide rule used for teaching but a 45 meter drum shaped slide rule would be awesome.
I never got to use a slider ruler but I got a couple of gems from my older coworkers that were catalogue slide ruler (not sure what its called). There was one that you pulled a couple sleeves and it gave you the clearance hole for the corresponding screw. It's kinda crazy how functional these manual lookup tables were and designing it must of been a form of art.
I've been reading on Smith Charts recently. It makes me wonder how people got anything done without the ability to do something as basic as complex arithmetic easily.
Just an anecdote, but I've got a Thacher's Cylindrical Slide Rule, manufactured around 1914 (based on SN). For comparison with the one in the article, the Thacher rule has a scale length of 18m (59 feet).
I found it on the scrap pile when an old building on my university was being renovated. It didn't have a university inventory serial number on it, so they weren't allowed to surplus it and had to toss it. I was in utter shock that it was going to get tossed, so took it.
The building it was in was built in 1913, originally to hold the School of Mines for Oregon State University, so I imagine it used to belong to a professor or researcher there when the School was quite new.
I've never actually used it, but perhaps some day can find a the original manual to learn how.
If you 'Unwrap' a Thacher's it ends up being the equivalent of a regular slide rule that's 59 feet / 18 meters long.
Here's one in a collection with a picture and some background...
I have a favorite circular slide rule: the E6B Flight Computor (sic). Pilots still use them to calculate airspeeds, distances, and fuel burn; and the reverse side can be used to calculate course corrections in the face of wind. Actually a lot of pilots probably use the iPad they keep all their air charts on for this, but it's great to have as a backup. All those flight parameters can be found with a simple mechanism that's made of card stock and uses no electricity! Sometimes the most marvelous tech is not high tech.
I remember seeing a small exhibit of slide rules including some cylindrical ones at MIT in the early 70's. It was in a few display cases in one of the older buildings (I think it was building 1, but it could have been building 5--they are connected.) We all had slide rules back then, mine was a bright yellow model made by Pickett and made of aluminum--it never operated as smoothly as the ones made of bamboo by K&E
I bought my slide rule in 7th or 8th grade with money I made from my paper route around 1964; I still have it and it functions just as well as when it was new.
I mentioned that we all had them, this was because there was really no alternative. Slide rules could perform all sorts of calculations, multiplication, division, exponentials, trig functions, all to an accuracy of around 3 significant digits. Computers existed, but terminals (with few exceptions) did not. Any calculation with a real computer required getting to the comp center and punching cards (any mistake on their funny keyboards meant throwing the card away and starting over). After that one would have to submit the deck to an operator behind a glass window. In ten minutes to an hour you would get the first result of running your program produced on fan fold paper, usually with green and white tinted background on the paper. The printing was done with a line-printer, often in upper case only.
Naturally, the first few attempts ended up with syntax errors and it was easy to waste a lot of time on simple calculations so the 3 significant digits of an ordinary slide rule started to look good enough.
Of course, we all also had (right next to our collegiate desktop Websters dictionary) a copy of the CRC Standard Mathematical Tables. This book, hundreds of pages long, had tables of logarithms and trig functions accurately to 5 digits which could solve these problems to 5 or 6 digits accuracy with careful interpolation techniques (which were taught in high school back then).
A year or two after I bought my slide rule I saw an early black and white episode of Lost in Space, a 60's TV show about a family lost on a remote planet. On the program the character Will Robinson was using a handheld calculator. It was pure science fiction device about the size of large 6cm thick hardback book. It contained a keyboard of perhaps 16 keys and a large, say 5cm, high display. I thought that it was so incredible, like the space suits and robots that appeared in the show. I wished that there was someway in the distant future that I would ever have such an amazing device!
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[ 3.5 ms ] story [ 99.7 ms ] threadOn a mildly related note, does anyone know if it's possible to construct a logarithmic scale from simple tools? A demonstration I've always wanted to try is building a slide rule "from scratch"
EDIT: by "simple tools" I mean no computers. Pen and paper, compass and straight edge, that sort of thing. I would assume in real life it was made via trial and error on a very large scale and shrunk down optically for a screen print, but I wonder if there's a "precise" way
After all, the slide rule was invented hundreds of years before the computer.
Compass and straight edges compute what are called the constructible numbers [1], which have 0, 1, and anything obtainable via addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers.
These will not allow you to get to logarithms. You can't avoid having to calculate them numerically; from there you may also create approximations to construct them but you might as well just use a ruler at that point.
You can compute logarithms by hand but they're notoriously tedious even by 19th century standards (and they had a great deal more tolerance for that sort of thing than we do today, for obvious reasons), which is why you could buy books full of them, and that's generally what people used.
[1]: https://en.wikipedia.org/wiki/Constructible_number#Algebraic...
Yeah, I don't know how practical a construction would be overall for someone who seriously went at it; I was mainly moved to answer because this objection that constructions can't express transcendental numbers just doesn't seem relevant -- digitally you don't keep infinite precision either.
Square roots and multiplying are two of the simplest geometric constructions (geometric mean and taking a proportion), compass and straightedge operations can be accurate, you can construct it enlarged and then scale it down, and finally the precision you need to aim for at the end is bounded. A potentially fun project idea. Think of it as a kind of retrocomputing: how might Euclid or Archimedes have designed a slide rule?
https://en.wikipedia.org/wiki/Pantograph
Log scale is a linear fractal, self-similar.
Say you constructed a pantograph where the ratio of the "inner" point to the "outer" point was ... um ... log(5)? for a decimal scale? ( I think you could make a binary scale but I can't think what the ratio would be log2(1)? )
Anyway, you'd start with the pantograph's base and outer point at the ends of your ruler, mark off the inner point, and then repeat on each sub interval, and then again recursively until you ran out of room.
Like I said, I could be wrong about this.
Well, there were also 5 computers (one analog) onboard the spacecraft, multiple powerful IBM System/360 mainframes computing trajectories on the ground, a Honeywell 1800 assembling code, a whole pile of Univac computers (1230, 494, 495) processing data, and RCA 110A computers for mission control (including one inside the launch platform under the rocket).
There are other softer benefits too, such as making the quantities in the computation somehow more... visceral.
Wikipedia has an article: https://en.wikipedia.org/wiki/E6B
Indeed the whole notion of logarithms was developed and introduced by Napier to simplify calculations!
If you pick some given resolution limit, it could all be done by hand, but would take quite a while.
Relevant articles:
https://sliderules.lovett.com/cookiedev/extendeddisplayartic...
https://sliderules.lovett.com/cookiedev/extendeddisplayartic...
Also, not actual a logarithm, but a wicked clever hack:
https://en.wikipedia.org/wiki/Irish_logarithm
To build a cardboard slide rule get a table of logarithms accurate to around 3 digits and layout a "ruler" with marks for the values from 1.0, 1.1, 1.2, ... 10.0. These marks however are placed on the "ruler" at the location corresponding to the log of the values: 1.0 is placed at the start of the ruler since log(1.0) == 0 and 10.0 is placed at the end of the ruler since log(10.0) == 1. In between the over values are placed where their log says they should go, so for example log(5.0) == 0.699 so the 5.0 should appear on the ruler 69.9 percent of the way from the 1.0 mark to the 10.0 mark.
Two of strips of cardboard labeled in this fashion will give you a very basic slide rule. Placing them next to each other it is very easy to position the slides so that you are "adding" the logs of two numbers. This is how multiplication is done using a slide rule.
Coincidentally, there's one for sale on eBay right now.
https://www.sliderulemuseum.com/Manuals/KL-1_RussianCircular...
(I now need reading glasses to use it- so annoying).
I was disappointed
Most of them are pretty cheap on Ebay, <$200 not including shipping. The Addiators in particular are simple, cheap and small.
I found it on the scrap pile when an old building on my university was being renovated. It didn't have a university inventory serial number on it, so they weren't allowed to surplus it and had to toss it. I was in utter shock that it was going to get tossed, so took it.
The building it was in was built in 1913, originally to hold the School of Mines for Oregon State University, so I imagine it used to belong to a professor or researcher there when the School was quite new.
I've never actually used it, but perhaps some day can find a the original manual to learn how.
If you 'Unwrap' a Thacher's it ends up being the equivalent of a regular slide rule that's 59 feet / 18 meters long.
Here's one in a collection with a picture and some background...
https://americanhistory.si.edu/collections/search/object/nma...
I bought my slide rule in 7th or 8th grade with money I made from my paper route around 1964; I still have it and it functions just as well as when it was new.
I mentioned that we all had them, this was because there was really no alternative. Slide rules could perform all sorts of calculations, multiplication, division, exponentials, trig functions, all to an accuracy of around 3 significant digits. Computers existed, but terminals (with few exceptions) did not. Any calculation with a real computer required getting to the comp center and punching cards (any mistake on their funny keyboards meant throwing the card away and starting over). After that one would have to submit the deck to an operator behind a glass window. In ten minutes to an hour you would get the first result of running your program produced on fan fold paper, usually with green and white tinted background on the paper. The printing was done with a line-printer, often in upper case only. Naturally, the first few attempts ended up with syntax errors and it was easy to waste a lot of time on simple calculations so the 3 significant digits of an ordinary slide rule started to look good enough.
Of course, we all also had (right next to our collegiate desktop Websters dictionary) a copy of the CRC Standard Mathematical Tables. This book, hundreds of pages long, had tables of logarithms and trig functions accurately to 5 digits which could solve these problems to 5 or 6 digits accuracy with careful interpolation techniques (which were taught in high school back then).
A year or two after I bought my slide rule I saw an early black and white episode of Lost in Space, a 60's TV show about a family lost on a remote planet. On the program the character Will Robinson was using a handheld calculator. It was pure science fiction device about the size of large 6cm thick hardback book. It contained a keyboard of perhaps 16 keys and a large, say 5cm, high display. I thought that it was so incredible, like the space suits and robots that appeared in the show. I wished that there was someway in the distant future that I would ever have such an amazing device!