One day while in his library in
Eratosthenes read that in the frontier outpost town of Syene at noon on
June 21st, vertical sticks cast no shadow while a reflection of the sun
could be seen at the bottom of the well. Being the scientist that he
he wanted to know if the same thing happened in Alexandria. He
that there were shadows there.
the "sun" angle is the same as the central angle!!!
But how does knowing this help in
the central angle? Eratosthenes used a little geometry to come up with
his insight. This discovery makes possible the use of the measure of
central angle (along with the distance between the two towns) to
the circumference of the earth.
Eratosthenes knew that the sun
enough away so that its rays are parallel. In the diagram above note
there is no shadow at Syene while there is a shadow in Alexandria. Then
he realized the sun angle had to equal the central angle. See the
there IS a shadow at both sites? Below is a model (created with the
Sketchpad) of the problem. Notice that neither sun angle A at
nor angle S at Syene equals the central angle.