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It is a widely known fact that any arbitrary angle cannot be constructed using the ancient Greek method of only using a compass and an unmarked straightedge. However, between then and now, we have invented countless instruments to construct and measure any arbitrary angle to desired precision, protractors being one of the earliest and the most popular. But for this development, humans must have figured out how to make a 1° angle reference, without an existing reference. I'm assuming it was similar to other methods used to circumvent other problems unsolvable by Greek methods, such as Neusis construction for the trisection of an angle. What was this method for a 1° angle? I cannot find any sources that would detail this historical aspect of angle measurement or even a modern method that isn't just "use a protractor". Any direction towards the answer would be helpful.

zwim
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DMa
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    I’m voting to close this question because it belongs to the History of Science and Mathematics StackExchange site. – José Carlos Santos Mar 13 '21 at 11:22
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    If you want some light amusement, here is how we calculated the sine of $1^\circ$on ProofWiki: https://proofwiki.org/wiki/Sine_of_1_Degree -- the cosine can be calculated thence, and from there $1^\circ$ can be constructed formally. – Prime Mover Mar 13 '21 at 11:23
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    I agree that this belongs on HSM.SE for documented methods. That said, without classical straightedge-and-compass pedantry, it's "easy" to solve this problem: Wrap a piece of rope around a wheel (or a disk of whatever size you like), unwrap it, lay it straight, mark whatever units you like, then wrap it back around the wheel and transfer your marks thereto. Effectively, you've created a protractor, so "use a protractor" is a perfectly sound and practical strategy for anyone wanting a $1^\circ$ angle. Indeed, I'd expect this method to be far more accurate than any multi-step "construction". – Blue Mar 13 '21 at 11:35
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    Note that neusis was not something to circumvent previously existing problems, but rather on the contrary, the Greeks deliberately dropped neusis from the set of valid tools and thereby introduced these problems in the first place. – Hagen von Eitzen Mar 13 '21 at 11:39
  • @Blue sorry I wasn't aware of HSM.SE. But from a non-historical perspective, as an evergreen method, possibly yours is how it went about. But given multi-step constructions would be practically inaccurate, theoretically they give exact 30°, 60°, 90°, etc. While the method you suggest has an error of the transformation from a straight line to a circle. Is there no exact theoretical method, using any array of tools (not limited to the Greek pedantry), to construct an angle of 1°? I feel this would be an entirely separate question from what I asked, and I'll move it if that's the case. – DMa Mar 14 '21 at 13:25
  • @PrimeMover that is a really interesting way to go about it. – DMa Mar 14 '21 at 13:31
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    @DMa: "the method you suggest has an error of the transformation from a straight line to a circle" It's theoretically possibly to accurately unwrap a circle (that's how we get involutes), mark units, and wrap it back up, but no matter. .. "Is there no exact theoretical method, using any array of tools (not limited to the Greek pedantry), to construct an angle of 1°?" Sure: Construct $3^\circ$ (via sums, products, and square roots, and use the neusis (or other means) to trisect. – Blue Mar 14 '21 at 14:08
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    @Blue I think that might be the exact answer I was looking for, thanks a lot. Would mark this as solved if the question wasn't closed :) – DMa Mar 14 '21 at 14:11

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