“How do we know the world is round” is an interesting example to use. Eratosthenes got an estimate for the radius of the Earth that was pretty close using, I dunno, sticks and his eyeball or whatever ancient Greeks had.

The history is useful there, I think. First off, it is motivating — that’s a pretty big question, really shows what math can accomplish without much hardware. Second, it provides some sort of useful constraints if you are trying to work out the computation yourself. In a textbook the problems are always a bit artificial, some info will be given, some hidden. “Anything an Ancient Greek would have” is a bit more real-world, you can see where the constraints come from.