Hello igottaquestion Originally Posted by

**igottaquestion** Around 255 BC Eratosthenes, a Greek mathematician, estimated the circumference of the earth. He discovered that on a certain day, in the city of Syene, one could see the sun's reflection at the bottom of a deep well. At the same time of day, 500 miles (measured along the earth's surface) North at Alexandria, the sun's rays formed an angle, Am of 7.5 degrees with the tower. Using this information, he was able to estimate the circumference of the earth. Explain how this could be done and estimate the circumference of the earth. How does this compare to the recent calculations of the circumference as 24,901.55 miles? (He worked as if the sun's rays were parallel.

You need to realise that a line that is 'vertical' at the point where you happen to be on the earth's surface is a line that passes through the centre of the earth - because that's the direction that gravity is pulling you (and a 'vertical' plumb line).

So if we assume that the well shaft and the tower are both 'vertical' at their own particular point on the earth's surface, then because the sun's rays are parallel, these two 'vertical' lines in fact make an angle of $\displaystyle 7.5^o$ with each other. So that's the angle these lines make when they meet at the centre of the earth.

The section of the earth's surface between the well and the tower (all $\displaystyle 500$ miles of it) is therefore $\displaystyle \frac{7.5}{360}$ of the whole circumference of the earth. So, if the circumference of the earth is C, we get:

$\displaystyle 500 =\frac{7.5}{360}C$

$\displaystyle \Rightarrow C = \frac{500\times360}{7.5}=24000$

So the circumference of the earth is approximately $\displaystyle 24000 $ miles, which is pretty close to today's measurement.

Grandad