The Chapel Hill News

February 15, 2004

Ask the Postdoc


By Phillip Manning


Dear Postdoc,
My reading of the National Geographic Atlas says you saw the sun set over the Gulf of Mexico ... [not] the Caribbean. How far is it to the horizon over water anyway? Thanks, CL

Dear CL: You refer to the Postdoc’s recent column in which he recalled seeing the green flash in Key West “as the sun slipped into the Caribbean.” Although some quibblers might rebut your argument by pointing out that the Gulf of Mexico is an arm of the Caribbean, the Postdoc accepts your correction. Your attention to detail makes me suspect that you are associated with an institution of higher learning. The Postdoc heard a similar chiding tone when he gave a seminar at such an institution while wearing a T-shirt bearing the message, “When All Else Fails, Manipulate the Data.” In fact, that T-shirt may account for the Postdoc currently being between opportunities, which is why he is available to answer science questions.
Now, on to your query. One can calculate the distance to the horizon using simple geometry. Consider a person, lets call him Pythagorus, standing on a seashore. Pythagorus is a short guy, around 5 feet 3 inches tall. That puts his eyes about 5 feet above the sea. Now, imagine a line that extends from Pythagorus’s eyes to the horizon, which we will call point B. Next, imagine a line running from the center of the Earth to Pythagorus’s eyes and another running from Earth’s center to point B, the horizon that Pythagorus sees. With a little more imagination, you can see that the three lines form a right triangle. And as we learned in high school, if you know the length of two sides of a right triangle, you can calculate the third side using Pythagorus’s famous theorem. The result is a simple formula stating that the distance to the horizon is 1.2 times the square root of the height of the eyes (in feet) looking at the horizon. Thus, in Pythagorus’s case, the horizon is 2.7 miles away.
Pythagorus, however, could not make this calculation in 500 B.C. because he didn’t know the Earth’s radius, which is one side of the right triangle. (The other side is the radius plus the distance of Pythagorus’s eyes above the water.) The radius of the Earth was measured three centuries later by another smart Greek geographer named Eratosthenes, who was director of the world-class library in Alexandria.
Eratosthenes heard that in the town of Syene, which is south of Alexandria, a vertical stick cast no shadow at noon on the day of the summer solstice. Being of scientific bent, he wondered if the same thing was true in Alexandria. So, he drove a stake into the ground at noon during the next summer solstice and found it did cast a shadow. Eratosthenes measured the angle of the shadow. Then, using not-so-simple geometry, he calculated that the distance from Alexandria to Syene was one-fiftieth of the Earth’s circumference. Now, he needed only to know the distance between the two towns to determine the Earth’s circumference.
To find the distance without accurate maps, Eratosthenes did what any high-ranking academic would do: He instructed a graduate student to pace off the distance between Syene and Alexandria. A few months later, the exhausted student (who remains anonymous because didn’t get his name on the paper) reported back. It was, he said, 5,000 stadia, a common unit of measurement in those days. From this, Eratosthenes calculated the circumference of the Earth to be 250,000 stadia, giving it a radius of 39,789 stadia.
Scholars have long tried to determine just how close Eratosthenes was to the true value of the Earth’s radius of 3,963 miles. Unfortunately, no one knows exactly how long a Greek stadium was. But applying a conversion factor calculated from ancient writings, one historian computed Eratosthenes’ value for the radius of the Earth to be 3,907 miles, astoundingly close to the modern value.
Eratosthenes went on to a distinguished career, estimating the distance to the moon and making other scientific contributions. No word has filtered down about the fate of the grad student who walked nearly a thousand miles round trip to help answer CL’s question. Nor do we know how he managed to get a result that produced an answer so close to the correct value. Did he really pace off those miles or did he somehow guess the answer in advance? We’ll never know. But I can see him now, rewarded with a job as assistant to the Great Man, shelving papyrus tomes the library, wearing a white toga carrying the message, “If all else fails ....”
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