Reflections 9/27/96
Robert Kern Curtis

The Size of the Earth

How big is the earth? What's its diameter? circumference? Who figured this out? How?

We go back to Alexandria, in Egypt, for this. The city Alexander the Great founded to be a library for the world. The head librarian, Eratosthenes, gets credit for measuring the earth. He knew that at a city 5000 stadia south of Alexandria along the Nile, the sundials cast no shadows on June 21st at noon. This city was named Syene and is now Aswan, the site where Colonel Nassar's government build a dam to tame the Nile. At Syene on June 21st, it was also true that the sun was seen reflected in the water at the bottom of the well at the center of town. All this meant that the sun was directly overhead on June 21st at noon. Today, we would expect this since Aswan is on the Tropic of Cancer and June 21st is the summer solstice.

A stadium was the distance once around the track in the stadium! We do not know exactly how long this distance was, but it seems to be about one-tenth of a mile; so the distance from Alexandria to Syene was 500 miles due south, as can be confirmed by inspecting a current map.

On June 21st, Erastosthenes measured the angle the sun made with the vertical at noon at Alexandria, and found it to be 7.2 degrees. This means that, because given two parrallel lines cut by a transversal alternate interior angles are congruent, the arc on the circumference of the earth between Syene and Alexandria is 7.2 degrees, and therefore the distance between the cities is 7.2/360 of the circumference of the earth. Thus the circumference of the earth is 25,000 miles. Using the definition of pi, the diameter of the earth is 8,000 miles.

Eratosthenes' results were so good that most scholars consider him to have been not only clever but also lucky. The fact that this calculation was done several centuries B.C. casts a shadow on Christopher Columbus's problems estimating the distance from Portugal to India and his sailors' worrying about falling off a flat earth. But, then, history books are never to be trusted anyway--don't they usually say that Roemer measured the speed of light?

-rkc

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