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"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 obt
by 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 agoOTOH 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 agoThe 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 agoYou'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 agoWell, if you prove me wrong and post it to HN I guarantee an upvote. :)
- abecedarius 6y agoIt's a dare, eh? :) I'll put it on the list.