Eratosthenes' Geodesy

The essential principle behind Eratosthenes’ reasoning can be understood by anyone with an elementary knowledge of geometry. On the scale of difficulty seen in the math section of a typical college entrance exam., the problem probably would rank around the middle. But like many innovations, the solution is remarkable not for its inherent difficulty, but because it took one remarkable mind to apprehend its power.

If the sun is directly overhead at Syene on the solstice, then the simultaneous angle of the sun’s shadow back at Alexandria, α, must by definition be same as the angle β between two lines drawn from the earth’s center to Syene and Alexandria. If angle β is known, and we know that the complete circumference of the circle contains 360 degrees (a convention borrowed by the Greeks from the Babylonians), then we can set up a simple proposition:

Angle β / 360 = Linear distance from Syene to Alexandria / Circumference of the earth

Eratosthenes took the distance from Syene to Alexandria to be 5,040 stades (a stade was an ancient unit of measure approximately equal to 600 modern feet). Using some vertically plump object at Alexandria on the day of the solstice, he measured the angle α, and therefore angle β, at 7.2 degrees. Knowing three of the terms in the quation above, he was then in a position to calculate the final term, the circumference of the earth. His final result, 252,000 stades, works out to 24,662.2 miles – just under 1 percent shy of the modern polar measurement of about 24,859.8 miles.