8 ms·

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.

jedimastert 6y ago
Sorry, I meant "from scratch" meaning no computers. Pen and paper, compass and straight edge, that kind of thing.
- Robotbeat 6y ago
  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.
  - kd5bjo 6y ago
    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|
- jerf 6y ago
  "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_definitions https://en.wikipedia.org/wiki/Constructible_number#Algebraic...
  - abecedarius 6y ago
    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.
    - jerf 6y ago
      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.
      - abecedarius 6y ago
        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?
        jerf 6y ago
        Well, if you prove me wrong and post it to HN I guarantee an upvote. :)
        abecedarius 6y ago
        It's a dare, eh? :) I'll put it on the list.
- carapace 6y ago
  Use a pantograph? https://en.wikipedia.org/wiki/Pantograph https://en.wikipedia.org/wiki/Pantograph Log scale is a linear fractal, self-similar.
  - jedimastert 6y ago
    That would be useful for geometric scales, which I don't believe a logarithmic scale is
    - carapace 6y ago
      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.
JKCalhoun 6y ago
Or take it to the next level: https://hackaday.com/2020/11/11/homebrew-slide-rule-gets-back-to-mathematical-basics/ https://hackaday.com/2020/11/11/homebrew-slide-rule-gets-bac...

jedimastert 6y ago

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

Robotbeat 6y ago
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.
- kd5bjo 6y ago
  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|
jerf 6y ago
"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_definitions https://en.wikipedia.org/wiki/Constructible_number#Algebraic...
- abecedarius 6y ago
  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.
  - jerf 6y ago
    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.
    - abecedarius 6y ago
      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?
      - jerf 6y ago
        Well, if you prove me wrong and post it to HN I guarantee an upvote. :)
        abecedarius 6y ago
        It's a dare, eh? :) I'll put it on the list.
carapace 6y ago
Use a pantograph? https://en.wikipedia.org/wiki/Pantograph https://en.wikipedia.org/wiki/Pantograph Log scale is a linear fractal, self-similar.
- jedimastert 6y ago
  That would be useful for geometric scales, which I don't believe a logarithmic scale is
  - carapace 6y ago
    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.

JKCalhoun 6y ago

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