We see a lot of papers and talk about ancient Babylonians exactness of calculating the value of square root of 2. For example: http://johncarlosbaez.wordpress.com/2011/12/02/babylon-and-the-square-root-of-2/
But how close could they approximate the value of SQRT3? I have a document that talks about the subject, but I cannot trace the answer directly from it: http://www.helsinki.fi/~whiting/roots.pdf
Maybe someone has knowledge of the method they used and reference to the old Babylonian clay tablets, where it can be verified and seen in use.