How Eratosthenes measured the circumference of the earth

Eratosthenes was this ancient greek guy who lived around the late 200s BC. He did a lot of work with Geography.

His claim to fame is that at one point, using nothing but simple geometric theorems and a stick, he managed to find a fairly accurate measurement of the circumference of the earth.

He got the idea when he was in Alexandria, where he heard of a deep well in Syrene (a town in southern Egypt) that for one day a year, at noon, was completely filled by light, with no shadows. He interpreted this correctly to mean that at that moment on that day, the sun was directly over the Syrene well.

So, when the day the Syrene well would be shadowless came, Eratosthenes (in Alexandria) went outside, stuck a stick straight vertically into the ground, and took the following logic:

• Assume the sun's rays are parallel to each other when they hit the earth.
• Assume the earth is round.
• Alternate interior angles of a line bisecting two parallel lines are equal. In other words, look at an N: the two angles on the inside bends of that N are equal to each other.

• |   |
  A   B   A is the top of the stick
  |\  |   B is the well the sun's ray hits head on
  | \ |   C is where the stick's shadow hits the earth
  C  \|   D is the center of the earth.
      D
  
  BD and AC are the sun's parallel rays
  AD is the stick
  ∠CAD ≡ ∠BDA

• ∠CAD was, on measurement, discovered to be 7.2°; an angle of 7.2 degrees makes a sweep of 1/50 of a circle's circumference.
• Therefore the distance from A to B (Alexandria to Syrene, 5000 stadia) is 1/50 of the circumference of the earth.
• Therefore the circumference of the earth is 250,000 stadia, or 46,250 km.

numbers from http://youth.net/eratosthenes/welcome.html