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Comment by jedimastert | Hacker News Reader

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

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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.

> [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!

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.

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.

Using straight edge and compass:

  - draw a line
  - pick two points on the line
  - label them 1 and 10
  - bisect the segment to get √10
  - bisect the halves to get 10^¼ and 10^¾
  - etc.

You can’t do much better, as the logarithms of rational numbers tend to be transcendental.

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.

Relevant articles:

https://sliderules.lovett.com/cookiedev/extendeddisplayartic...

Also, not actual a logarithm, but a wicked clever hack:

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

Are slide rules still useful? I mean, are there calculations that are more convenient to do with a slide rule than a calculator?

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.

Wikipedia has an article: https://en.wikipedia.org/wiki/E6B

You can plot the lines using matplotlib, then print it and stick it on a piece of wood. There are also slide rule PDFs online.

Sorry, I meant "from scratch" meaning no computers. Pen and paper, compass and straight edge, that kind of thing.

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):

  |        10         |  => divide by log10(1/2) : 1
  |    5    |    5    |  => divide by log10(1/5) : 1
  |1|   4   |1|   4   |  => divide by log10(1/2) : 1
  |1| 2 | 2 |1| 2 | 2 |  => divide by log10(1/2) : 1   
  |1|1|1|1|1|1|1|1|1|1|

"Pen and paper, compass and straight edge,"

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...

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?

Well, if you prove me wrong and post it to HN I guarantee an upvote. :)

It's a dare, eh? :) I'll put it on the list.

Use a pantograph?

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

Log scale is a linear fractal, self-similar.

That would be useful for geometric scales, which I don't believe a logarithmic scale is

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.

Like I said, I could be wrong about this.

Or take it to the next level: https://hackaday.com/2020/11/11/homebrew-slide-rule-gets-bac...

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