Circumference of the Earth Problem

Circumference of the Earth Problem

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At noon on the summer solstice in the Egyptian town of Aswan, the sun is straight overhead: objects cast no shadow and sunlight falls directly down a deep well. When he first read this fact, Eratosthenes, the librarian at Alexandria in the third century B.C., realized that he had the information he needed to estimate the circumference of the Earth.

At noon on the summer solstice, he measured shadows in Alexandria, finding that the solar rays there had a bit of a slant. The solar rays in Alexandria deviated from the vertical by seven degrees. Thus, the angle of elevation of the Sun in Aswan was 90 degrees above the horizon, and the angle of elevation of the Sun in Alexandria was 83 degrees above the horizon. He also knew that the two towns were about 5,000 “stadia” apart.

How did Eratosthenes discover the circumference of the Earth, and what was his answer? Give your answer in stadia. A stadia is the length of a typical Greek stadium. Although scholars disagree over the length of a Greek stadium, Eratosthenes’ logic is faultless given the instruments of his time. Some scholars think that his answer was correct within 5% of the actual value of the Earth’s circumference.

(Hint: Because the Sun is so far away from the Earth, assume that all of the Sun’s rays are parallel no matter where they strike the Earth.)