Eratosthenes’ Circumference

by James Buckland

NB: This post was published at 12:00pm on June 21st, 2013, the summer solstice.

Due to the Earth’s tilt on its axis, (23.4˚) the Sun traces out a variable path in the sky: its arc changes from day to day, depending on the season, as well as where on Earth you are. If you’re in a polar region, like Alaska, the Sun will stay very low in the sky, close to the horizon. If you’re in an equatorial region, like Egypt, the Sun will stay very high in the sky, far from the horizon, but close to the zenith — the point directly overhead, a vector normal to the surface of the Earth at your location.

If there were no axial tilt, the Sun would appear directly overhead at noon to anybody on the equator, all the time. Instead, people on the Tropic of Cancer (~23˚ north of the equator) can observe the Sun at the zenith during the summer solstice, and people on the Tropic of Capricorn (~23˚ south of the equator) can observe the Sun at the zenith during the winter solstice.

earthcir

Armed with this knowledge, Eratosthenes, a Greek mathematician and geographer working in Alexandria, Egypt, around 200 BC, was able to estimate to within 16% accuracy the circumference of the Earth. He knew that the city of Swenet was directly on the Tropic of Cancer, as confirmed by anecdotal evidence that, on the summer solstice, a man’s shadow could not be seen, even in a deep well. From the city of Alexandria, ~5000 stadia due north of Swenet, he use a sundial to measure that, on the summer solstice, the Sun was 7˚12′ south of the zenith, around 1/50th of a circle. Thus, the distance from Alexandria to Swenet must be about 1/50th of a circle. Depending on the measured value of a stadia (we can assume the Egyptian stadia of ~157.5m), he found the distance between the cities to be ~788km, which puts the circumference of the earth at 39 400 km, just 1.7% shy of the true value of 40 075 km, which is absolutely phenomenal.

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