Eratosthenes' method for determining the size of the Earth was an elegant application of simple geometry to an otherwise very difficult problem. By using the difference in the elevation of the noontime sun at two different locations, he was able to measure the angular difference between the vertical directions at those two locations. This angular difference told him what fraction of the way around the earth separated the two locations. He then used this fraction and the measured distance between the two locations to estimate the distance around the earth (a.k.a. the circumference).
As the story goes, old Eratosthenes learned that at noon on one day a year (the summer solstice, but that's not really important), the Sun shone directly down a deep vertical well in a town called Syene, about 500 miles south of his home in Alexandria. Eratosthenes reasoned that the Sun must be directly overhead at noon in Syene at this time. He knew that the Sun was not directly overhead in Alexandria on this day, so he figured that the vertical direction in Alexandria was different from the vertical direction in Syene.