How did Eratosthenes measure the size of the Earth?

Eratosthenes, a Greek geographer who lived from about 276 to 194 B.C., used shadows and geometry. On one day a year the Sun shone straight down a well at Syene, so it was directly overhead. At the same moment in Alexandria, a vertical post still cast a shadow. The angle of that shadow showed what fraction of a full circle separated the two cities, and the distance between them could then be multiplied up to the whole Earth. [1] [2]

Eratosthenes heard that at noon on the summer solstice, around June 21, sunlight lit the entire bottom of a well in Syene, now Aswan, and cast no shadow. He took this to mean the Sun was directly overhead. In Alexandria, almost due north, he measured the shadow of a vertical stick or post at noon on the same day and found an angle of about 7.2 degrees, roughly one fiftieth of a circle. Some classroom sources round this to 7 degrees or 7.5 degrees. [2] [1] [3]

Because the Sun is so far away, its rays reach Earth almost parallel. The shadow angle in Alexandria therefore equals the angle between the two cities as seen from Earth's center. Since 7.2 degrees is one fiftieth of 360 degrees, the distance between the cities is one fiftieth of the circumference. Eratosthenes put that distance at 5000 stadia, so he multiplied by 50 to reach about 250,000 stadia. The method works for any great circle, not only along a line of longitude. [1] [4]

The distance was reportedly found by hired surveyors called bematists, who were trained to walk with equal steps. This was seen as more reliable than timing camel caravans. One Berkeley astronomy page instead says he estimated the distance from a camel's average speed, so accounts differ on exactly how the figure was obtained. [2] [5]

The result depends on the length of the stadium, which scholars do not agree on. Values between about 500 and 600 feet have been suggested, giving a circumference between roughly 24,000 and 29,000 miles. The Berkeley page gives a range of 39,690 to 46,620 km and says the result was within 16 percent of the true polar value of 40,008 km. NOAA gives 28,740 miles, or 45,984 km. NSTA uses a distance of 800 km, which gives about 40,000 km, very close to the modern figure. The modern value is about 24,900 miles around the equator, slightly less around the poles. [2] [5] [1] [4]

His assumptions were not exactly right. He took Alexandria to be due north of Syene and Syene to lie exactly on the Tropic of Cancer. These were close enough for a good measurement. His original manuscript is lost, and his method is known through later Greek writers. [2]