How Did Eratosthenes Come So Close?

By Joy Hakim

Syene, near modern Aswân, Egypt, is close to the tropic of Cancer (23.5° N). At that line of latitude, at noon on the summer solstice (June 20 or 21), the Sun is directly overhead. Stand there and you'll have no shadow. Alexandria is farther north, so there the noon Sun is not quite overhead on the solstice. You, or a stick, will cast a shadow that's at about a 7° angle.

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Both cities lie near the same meridian, or longitude line, which is a north-south great circle (a circumference, really) around the Earth. Like all circles, the meridian has 360°. Since 7° is about 1/50 of 360°, the distance between the two cities (5,000 stadia) must be 1/50 of the distance around the meridian. By multiplying 5,000 by 50, Eratosthenes estimated Earth's circumference at 250,000 stadia. Then, since he was aiming for an easy-to-use number, he added 2,000 stadia and got 252,000. (It's evenly divisible by 60 and 360.)

So, you might be wondering by now, just how long was a stadium (singular for stadia)? The experts aren't quite sure, but they say it's between 150 and 158 meters (164 and 173 yards). At 157 meters (172 yards), a popular choice, Eratosthenes figured Earth's circumference at about 39,250 kilometers (24,390 miles). That's amazingly close to today's measures of the north-south circumference, which is about 40,000 kilometers (24,855 miles).

Why Was It So Tough?

Today's satellites can map and measure the Earth within inches of accuracy. It wasn't that easy for Eratosthenes. He had no way to measure long distances well, and even very careful step-counting is an uneven thing. The round number 5,000 stadia is a clue that Eratosthenes knew the result was an estimate.

Also, Alexandria isn't exactly due north of Aswân/Syene. It's a couple of degrees west of it. Aswân/Syene isn't on the tropic of Cancer. It's a little north of it. Eratosthenes couldn't know about those slightly off-kilter positions, which threw off his result a bit.

There's one more thing no one knew back then: Earth bulges at the middle and is a little flattened at the poles. That makes the meridians (the north-south great circles) a little shorter than the east-west Equator, which is 40,075 kilometers (24,901 miles) long.

Those tiny quirks probably wouldn't have mattered to Eratosthenes. He fudged by tacking on extra stadia to make the math easier, which shows he wasn't aiming for perfection. He just wanted a close estimate, and he did incredibly well.

Joy Hakim, former teacher, reporter, and editor, writes nonfiction for children. Her latest project is a six-volume series called The Story of Science, from which this article is excerpted with permission. ©2004 The Story of Science, Smithsonian Books. Her 10-volume series, A History of US, won the Michener Prize in Writing and was made into a PBS special called "Freedom."

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