How Eratosthenes Measured the Size of Earth

The size of Earth was first measured in Egypt by Eratosthenes in about 235 bc. He calculated the circumference of Earth in the following way. He knew that the Sun is highest in the sky at noon on the day of the summer solstice (which occurs around June 21 on today’s calendars). At this time, a vertical stick casts its shortest shadow. If the Sun is directly overhead, a vertical stick casts no shadow at all. Eratosthenes learned from library information that the Sun was directly overhead at noon on the day of the summer solstice in Syene, a city south of Alexandria (where the Aswan Dam stands today). At this particular time, sunlight shines directly down a deep well in Syene and is reflected back up again. Eratosthenes reasoned that, if the Sun’s rays were extended into Earth at this point, they would pass through the center. Likewise, a vertical line extended into Earth at Alexandria (or anywhere else) would also pass through Earth’s center.
At noon on June 22, Eratosthenes measured the shadow cast by a vertical pillar in Alexandria and found it to be 1/8 the height of the pillar. This corresponds to a 7.1° angle between the Sun’s rays and the vertical pillar. Since 7.1° is 7.1/360, or about 1/50 of a circle, Eratosthenes reasoned that the distance between Alexandria and Syene must be 1/50 the circumference of Earth. Thus the circumference of Earth becomes 50 times the distance between these two cities. This distance, quite flat and frequently traveled, was measured by surveyors to be about 5000 stadia (800 kilometers). So Eratosthenes calculated Earth’s circumference to be 50 * 5000 stadia = 250,000 stadia. This is very close to the currently accepted value of Earth’s circumference.
We get the same result by bypassing degrees altogether and comparing the length of the shadow cast by the pillar to the height of the pillar. Geometrical reasoning shows, to a close approximation, that the ratio shadow length/pillar height is the same as the ratio distance between Alexandria and Syene/Earth’s radius. So, just as the pillar is 8 times taller than its shadow, the radius of Earth must be 8 times greater than the distance between Alexandria and Syene. Since the circumference of a circle is 2p times its radius (C = 2pr), Earth’s radius is simply its circumference divided by 2p. In modern units, Earth’s radius is 6370 kilometers and its circumference is 40,000 km.



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