[JohnnyZ]

(1351/780) > sqrt(3) > (265/153).

No-one knows how he computed this. Continued fractions?

It's kind of hard to imagine what kind of math he'd be working with. (Euclidean geometric proofs were easy because the rules were clearly laid out at the beginning.) It's easy today to see that:

780^2 * 3 + 1 => 1351^2 and thus is just a smidge greater than [sqrt(3)]^2

and similarly with 265/153.

But how Archimedes came up with it?

[Virginia-American]

Here's a discussion of one possibility. I've seen geometric versions, but didn't see any online.