Circumference Of The Earth 


Eratosthenes computed it by noticing that on a particular day of the year in Syene the Sun was directly overhead. Objects cast no shadows, and the Sun's reflection could be seen in a deep well.
On the same day in Alexandria, a shadow was cast at an angle of about 1/50 of a circle (7.2 degrees). That meant that the distance from Syene to Alexandria must be 1/50 of the circumference of the Earth. Making several assumptions about the unit he used, and allowing for the various errors, that means he might have got an answer of around 39,690 km, which is astonishingly accurate.
It's an interesting question as to how Eratosthenes knew the distance was 500 stade (or whatever). There are suggestions that he measured how fast a camel walks, and then rode a camel from Alexandria to Syene at a constant pace, day and night, and worked it out from that. Possible in theory, although hard to believe.
Whether he did or not, the method is sound. We can then use this to compute the other useful measure of the size of the Earth, namely, the radius of the Earth.