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Eratosthenes' Estimate

Eratosthenes' Estimate

Eratosthenes of Cyrene, in addition to his experience as a geographer, was also a mathematician, poet, philosopher, historian, philologist, and chronologist. Born around 275 B.C., he studied at Plato's school in Athens and later was appointed director of the library at Alexandria.

Eratosthenes's most famous calculation is the length of the circumference of Earth. On the day of summer solstice (the day of the year with the longest period of daylight), Eratosthenes observed that the sun's rays shone directly down into a well in Syene at noon. The city of Alexandria lies very close to due North of Syene, meaning the two cities experience noon at approximately the same time (this is a necessary condition for the experiment). Eratosthenes measured the angle that the sun's rays made with a vertical stick located in Alexandria at noon on summer solstice and found the angle to be one-fiftieth of a circle, or 7.2°. Using the fact that the sun's rays are virtually parallel, Eratosthenes realized that an angle of 7.2° whose vertex was located at the center of the earth created an arc from Syene to Alexandria. Eratosthenes further estimated that the length of this arc, or distance between the two cities, was approximately 5,000 stades (a stade is an ancient unit of length approximately equal to a tenth of a mile). Since this distance of 5,000 stades represented one-fiftieth of the circumference of Earth, Eratosthenes was able to estimate the circumference of Earth to be 50 times that amount, or about 250,000 stades. He later revised his estimate to 252,000 stades, perhaps to utilize the fact that this number is divisible by 60. 252,000 stades is roughly equivalent to 29,000 miles. The actual circumference of Earth is 24,902 miles, making Eratosthenes's estimate about 16% high. Two of Eratosthenes's assumptions combined to create this error in measurement. First, Alexandria lies about 3° west of North from Syene, meaning the two cities do not experience noon at exactly the same time. Second, the distance between Syene and Alexandria is about 4,530 stades rather than 5,000 stades. Nevertheless, Eratosthenes' estimate is remarkably close to the actual circumference of Earth.

The Greeks also used a fixed star to measure the circumference of Earth. The Greek philoshoper, Posidonuis, is largely credited with pioneering this technique. His estimate for the circumference of Earth was about 20,700 miles, which is 4,202 miles less than the actual circumference. Today, satellites and other sophisticated tools measure the circumference of Earth with a high degree of precision.

You can get some more information about the History of Mathematics in Greece from the Clark University Mathematics and Computer Science Department .