4.Explain how Eratosthenes calculated the circumference of the earth. How close did he get?

Answer 1

In 215 B.C., Eratosthenes realized the Earth was round, but he had no way to prove it. He based his theory on the fact that, during a lunar eclipse, the shadow of the Earth always has a circular edge as it passes across the Moon. The only shape that could cast such a shadow was a sphere or circle.
As one of the most intelligent mathematicians and astronomers of his time, he developed the following experiment to measure the diameter of the planet.
1. He chose two points on the globe, one at Alexandria, Egypt, the other at Syene, in Egypt. He knew the distance between the two points was 5000 stades, a unit of length used in early Greece, about 6 stades per 1 kilometer.
2. Eratosthenes placed a stick into the ground in Alexandria, ensuring that it was as perpendicular as possible to the center of the planet. His assistant, in Syene, did the same.
3. On the day of the Summer Solstice, in the month we call June, the Sun shone straight down a well shaft. At local noon time in Syene, when the Sun was at its zenith in the sky, the assistant measured the angle cast by the shadow of the stick on the ground. Since the Sun was directly overhead, there was virtually no shadow.
4. At the same time, in Alexandria, Eratosthenes measured the angle of the shadow cast by his stick. It was approximately 7 degrees.
5. Mathematicians of the time already had decided that a complete circle encompassed 360 total degrees around its circumference. Thus, Eratosthenes calculated that the distance from Alexandria to Syene, 5000 stades, was about 7/360 the circumference of the Earth, or fifty times greater than the distance between the two sticks. Therefore, Eratosthenes calculated the Earth's total circumference to be 50 times 5,000 equals 250,000 stades.

Converting this measurement to kilometers yields 250,000 stades divided 6 kilometers per stadia equals approximately 42,000 kilometers for the circumference of the Earth. This was remarkably accurate for the time, as our own modern measurements show a more precise 40,000 kilometer circumference.

Answer 2

There was a location where the sun shone directly down a well on the day of the summer solstice. On the same day, Erasthones measured the angle of the Sun in the sky in another location.

Since he knew the distance from the measuring location to the well, he used the angle as an arc representating a portion of the Earth's circumference. Then he did his calculation by using the angle and the distance. He came pretty close!