Notwithstanding the ignorance of DUmmies in real science and social evolution, Archimedes didn't figure out the size of the earth. That particular accomplishment goes to Eratosthenes. At this time, it was generally known among the Greeks that the earth was a sphere because of observations both locally (ships disappearing in the horizon consistently, no matter what direction it was observed from), and extraneously (movement of stars, moon, sun). However, there was no known way that they could figure out its size.
Eratosthenes, though, noticed that on the same day, a two identical-length sticks located 800 miles apart, on a direct north-south axis (determined by the North star, whose position doesn't alter from anwhere on the same latitude), cast different=length shadows. He reasoned that if the earth was a sphere, the size of the sphere could be determined by calculating the different lengths of the shadows cast at high noon in both locations on the same day. He calculated the earth to be 25,000 miles in circumference, which was remarkably close to it's actual circumference of 24,901 miles. (In fact, the southern city he measured cast no shadow, which made the calculation much easier.) This was the first time the earth was correctly calculated with hard scientific evidence.
Unfortunately, most people could not believe that the earth was that big, and they were using another figure that was bandied about was put at about 18,000 miles. When Ptolemy wrote his book, he calculated the circumference of the earth to be 18,000 miles, and this was the length used as gospel until about around the 1500's. Indeed, Christopher Columbus used this figure in getting financing for his trip west.
But the actual calculation was done in Greece during BC times.
