> 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

> 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.