More Precisely 1-3

MORE PRECISELY 1-3 Earth Dimensions
In about 200 b.c. a Greek philosopher named Eratosthenes (276—194 b.c.) used simple geometric reasoning to calculate the size of Earth. The logic he employed still provides the basis for all measurements of distance outside of our own solar system. Eratosthenes knew that at noon on the first day of summer, observers in the city of Syene (now called Aswan), in Egypt, saw the Sun pass directly overhead. This was evident from the fact that vertical objects cast no shadows and sunlight reached to the very bottoms of deep wells, as shown in the insets in the accompanying figure. However, at noon of the same day in Alexandria, a city 5000 stadia to the north, the Sun was seen to be displaced slightly from the vertical. (The stadium was a Greek unit of length, believed to have been about 0.16 km, although the exact value is uncertain—not all Greek stadia were the same size!) Using the simple technique of measuring the length of the shadow of a vertical stick and applying elementary trigonometry, Eratosthenes determined the angular displacement of the Sun from the vertical at Alexandria to be 7.2 °. What could have caused this discrepancy between the two measurements? It was not the result of measurement error—the same results were obtained every time the observations were repeated. Instead, as illustrated in the figure, the explanation is simply that Earth's surface is not flat but is actually curved. Our planet is a sphere. Eratosthenes was not the first person to realize that Earth is spherical—the philosopher Aristotle had done that over 100 years earlier (see Interlude 2-2)—but he was apparently the first to build on this knowledge, combining geometry with direct measurement to infer the size of our planet. Here's how he did it. Rays of light reaching Earth from a very distant object, such as the Sun, travel almost parallel to one another. Consequently, as shown in the figure, the angle measured at Alexandria between the Sun's rays and the vertical (that is, the line	joining Alexandria to the center of Earth) is equal to the angle between Syene and Alexandria, as seen from Earth's center. (For the sake of clarity, this angle has been exaggerated in the drawing.) The size of this angle in turn is proportional to the fraction of Earth's circumference that lies between Syene and Alexandria. Since 7.2 ° is 1/50 of a full circle (360 °), so Earth's entire circumference can be estimated by multiplying the distance between the two cities by a factor of 50. We can express this reasoning as follows: (Notice that this is precisely the same geometric reasoning as that presented in Section 1.5.) Earth's circumference is therefore 50 5000, or 250,000 stadia. If we take the stadium unit to be 0.16 km, we find that Eratosthenes estimated Earth's circumference to be about 40,000 km. Since the circumference C of a circle is related to its radius r by the relation C = 2r, it follows that Earth's radius is 250,000/2stadia, or 6366 km. The correct values for Earth's circumference and radius, now measured accurately by orbiting spacecraft, are 40,070 km and 6378 km, respectively. Eratosthenes' reasoning was a remarkable accomplishment. More than 20 centuries ago, he estimated the circumference of Earth to within 1 percent accuracy, using only simple geometry. Even if our modern value for the size of one stadium turns out to be incorrect, the real achievement—that a person making measurements on only a small portion of Earth's surface was able to compute the size of the entire planet on the basis of observation and logic—is undiminished.

MORE PRECISELY 1-3 Earth Dimensions

In about 200 b.c. a Greek philosopher named Eratosthenes (276—194 b.c.) used simple geometric reasoning to calculate the size of Earth. The logic he employed still provides the basis for all measurements of distance outside of our own solar system. Eratosthenes knew that at noon on the first day of summer, observers in the city of Syene (now called Aswan), in Egypt, saw the Sun pass directly overhead. This was evident from the fact that vertical objects cast no shadows and sunlight reached to the very bottoms of deep wells, as shown in the insets in the accompanying figure. However, at noon of the same day in Alexandria, a city 5000 stadia to the north, the Sun was seen to be displaced slightly from the vertical. (The stadium was a Greek unit of length, believed to have been about 0.16 km, although the exact value is uncertain—not all Greek stadia were the same size!) Using the simple technique of measuring the length of the shadow of a vertical stick and applying elementary trigonometry, Eratosthenes determined the angular displacement of the Sun from the vertical at Alexandria to be 7.2 °.

What could have caused this discrepancy between the two measurements? It was not the result of measurement error—the same results were obtained every time the observations were repeated. Instead, as illustrated in the figure, the explanation is simply that Earth's surface is not flat but is actually curved. Our planet is a sphere. Eratosthenes was not the first person to realize that Earth is spherical—the philosopher Aristotle had done that over 100 years earlier (see Interlude 2-2)—but he was apparently the first to build on this knowledge, combining geometry with direct measurement to infer the size of our planet. Here's how he did it.

Rays of light reaching Earth from a very distant object, such as the Sun, travel almost parallel to one another. Consequently, as shown in the figure, the angle measured at Alexandria between the Sun's rays and the vertical (that is, the line

joining Alexandria to the center of Earth) is equal to the angle between Syene and Alexandria, as seen from Earth's center. (For the sake of clarity, this angle has been exaggerated in the drawing.)

The size of this angle in turn is proportional to the fraction of Earth's circumference that lies between Syene and Alexandria. Since 7.2 ° is 1/50 of a full circle (360 °), so Earth's entire circumference can be estimated by multiplying the distance between the two cities by a factor of 50. We can express this reasoning as follows:

(Notice that this is precisely the same geometric reasoning as that presented in Section 1.5.) Earth's circumference is therefore 50 5000, or 250,000 stadia. If we take the stadium unit to be 0.16 km, we find that Eratosthenes estimated Earth's circumference to be about 40,000 km. Since the circumference C of a circle is related to its radius r by the relation C = 2r, it follows that Earth's radius is 250,000/2stadia, or 6366 km. The correct values for Earth's circumference and radius, now measured accurately by orbiting spacecraft, are 40,070 km and 6378 km, respectively.

Eratosthenes' reasoning was a remarkable accomplishment. More than 20 centuries ago, he estimated the circumference of Earth to within 1 percent accuracy, using only simple geometry. Even if our modern value for the size of one stadium turns out to be incorrect, the real achievement—that a person making measurements on only a small portion of Earth's surface was able to compute the size of the entire planet on the basis of observation and logic—is undiminished.