Eratosthenes measured the circumference of the Earth, which in turn lets us compute the radius of the Earth. The Earth is known to be (more-or-less) a sphere, because the shadow it casts on the Moon during an eclipse is always a portion of a circular arc. We can also use that arc to compute the relative sizes of the Earth and Moon, and then the value given by Eratosthenes lets us compute the size of the Moon.

We don't actually know exactly the figure that Eratosthenes got because we don't know how big a "stade" was.

The Arabic scholar al-Biruni in about 990AD calculated the Earth's circumference, getting an answer remarkably close to modern value. Previously the circumference was measured by sighting the Sun simultaneously from two different locations. Instead al-Biruni developed a new method using the angle between a plain and mountain top which made it possible for it to be measured by a single person from a single location.

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