One of Archimedesí many significant contributions to mathematics was his approximation of the value of pi.  He was the first mathematician to establish a theoretical calculation for pi instead of an estimation.  By inscribing and circumscribing polygons on a circle, he was able to constrain the value of pi between 3+10/71 and 3+1/7.

Rather than trying to measure the polygons individually, he used one of Euclidís theorems to develop a faster numerical procedure.  This process enables one to get a result as accurate as desired.  It is a unique idea because he eliminated the geometrical aspect of it and turned it into an arithmetic procedure.  Throughout Archimedes proof for this process, he makes references to several square roots.  He does not say where he got the approximations.  This premature ability to calculate irrational square roots is remarkable.