5 ms·

I agree it's absurd just say that the rational trig system is "more accurate" and leave it at that. The idea that we have, for thousands of years, been ignorin

I agree it's absurd just say that the rational trig system is "more accurate" and leave it at that. The idea that we have, for thousands of years, been ignoring mathematical wisdom of the ancient Babylonians is pure click-bait. But the issue goes beyond the maths itself and into what humans want to do with it. Let's start with an apparent nitpick. You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample. For modern tables, the samples tend to be regularly spaced, while the Babylonians apparently chose to sample the rational points. Which is better likely depends on the use case. It's not obvious that "normal trig" is better from an engineering point of view. At least not for all engineers. If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. That is, it's fairly easy to make rational ratios between the opposite. It's then handy to know what mathematical properties the resulting angle has. Or else to be able to look at your table and pick ratios that have some useful property. You could not do this with a "normal" trig table, or even a modern a hand calculator.

Lazare 9y ago
> You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample That's true, and my wording was poor. But during the heyday of trig tables, they were very thorough, giving you results for angles (typically) very, very close to whatever you were trying to calculate, and to usefully high number of decimal places. (I'm old enough - just - that I actually used them in school.) But more to the point, we don't use trig tables any more, we calculate the number on the fly to any arbitrary desired degree of accuracy. That's always going to be the best choice; there's no tradeoff here. > If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. Also quite true, but of course, no temple wall is exactly N cubits high; the measured height is at best an approximation, so the resulting ratio of height to width (or whatever) is at best an approximation. All we need from our trig tables (or calculator) is enough accuracy. And in practice, that's not hard. The reality is that, as the linked article says, "it seems like a solution to a problem that doesn’t exist". When we actually used trig tables to build bridges, we didn't have issues with trig tables, and now we have a much better solution. Trig tables that, as you put it, sample the rational points just doesn't seem like it's better for any use case.
- jacobolus 9y ago
  In modern times the best solution is to usually never bother with angle measure, and just stick to pure vectorial methods. If you happen to need an angle measure for whatever reason (e.g. interacting with legacy systems) you can then compute it to however many binary digits are appropriate for your needs.

Lazare 9y ago

> You say "rather than, as modern trig tables do, giving it for all angles". But no table gives all angles, it's always sample That's true, and my wording was poor. But during the heyday of trig tables, they were very thorough, giving you results for angles (typically) very, very close to whatever you were trying to calculate, and to usefully high number of decimal places. (I'm old enough - just - that I actually used them in school.) But more to the point, we don't use trig tables any more, we calculate the number on the fly to any arbitrary desired degree of accuracy. That's always going to be the best choice; there's no tradeoff here. > If you are designing a Babylonian temple then you have the freedom to make your rooms so-many-cubits by so-many. Also quite true, but of course, no temple wall is exactly N cubits high; the measured height is at best an approximation, so the resulting ratio of height to width (or whatever) is at best an approximation. All we need from our trig tables (or calculator) is enough accuracy. And in practice, that's not hard. The reality is that, as the linked article says, "it seems like a solution to a problem that doesn’t exist". When we actually used trig tables to build bridges, we didn't have issues with trig tables, and now we have a much better solution. Trig tables that, as you put it, sample the rational points just doesn't seem like it's better for any use case.

jacobolus 9y ago
In modern times the best solution is to usually never bother with angle measure, and just stick to pure vectorial methods. If you happen to need an angle measure for whatever reason (e.g. interacting with legacy systems) you can then compute it to however many binary digits are appropriate for your needs.