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**1. When the sun was directly overhead in syene, why was it not directly overhead in Alexandria**

The size of Earth was first measured in Egypt by Erathosthenes 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 June 22, the summer solistice. At this time, a vertical stick cast its shortest shadow. If the Sun is directly overhead a vertical stick casts no shadow at all, which occurs at the summer solstice in Syene, a city south of Alexandria (where the Aswan Dam stands today). Eratosthenes learned that the Sun was directly overhead at the summer solstice in Syene from library information, which reported that, 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 verical 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 of 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 Erastosthenes calculated Earth's circumference be 50 X 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 shadow cast by pillar to the height of 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 greater 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 2∏ times its radius ( C = 2∏r), Earth's radius is simply its circumference of a divided by 2∏. In modern units, Earth's radius is 6370 kilometers and its circumference is 40,000 km.

## Homework Equations

**3. because the angle 7.1° can correlate to the length of distance between alexandria and syene which is 7.1/360 which would be 1/50 of the earth but a different distance than syene I am sure on another day other than June 22 the sun would be directly over alexandria due to the fact of the Earth's angle and the sun and the earths rotation.**