
When we were calculating the circumference of the earth,

our calculation was based entirely on the assumption that the sun's rays struck

Alexandria and Syene parallel.

Let's examine just how true or how good that assumption was.

Now, the drawing I've drawn here is totally out of scale.

I've drawn the earth huge and only a section of it.

I've drawn the sun absurdly tiny and much, much too close to earth.

For the sake of the geometry, this picture will work.

Now, we know that the distance from the earth to the sun is about 150,000,000 km.

Eratosthenes knew that the distance from Syene to Alexandria.

This distance was about 5000 stadia.

Remember 1 stadion is equal to about 185 meters.

I want to know what's this angle.

The reason why I want to know what this angle is

is because this angle somehow quantifies how parallel these lines are.

If they're truly parallel, this angle would be zero degrees.

This line would be coming in at the exact same orientation as this line.

Any deviation from 0 represents some error,

and we want to make sure that if it does deviate from zero

that it's still quite a small number.

For this problem, you're going to want to assume that this is a right triangle,

even though it doesn't quite look like one.

If we drew this more to scale, you would see that it truly is very close to being a right triangle.

So, we can still use our trigonometry.

Using the fact that this distance is 150 million km,

which you could say is the length of this leg here, if you like,

or that leg, since these are equal.

You know this length is 5000 stadia.

Can you tell me what's this angle?

We'll call this angle α.

Give your answer in degrees.

For this problem, I actually want you to round to 5 decimal places.