
タイトル：
0144 Solving for Alpha

概説：

Now, we can go back to the original problemthe thing we've been trying to solve this whole unit.

As you remember we have the sun's rays striking perpendicular to the earth here at Syene.

We've seen that these rays all came in parallel like what's around here and here in Alexandria

which is north of Syene, the rays no longer strike the earth perpendicular to the surface of the earth.

They strike at this angle α.

We showed before that must be equal to this angle α as well.

And by comparing the full angular measure of the circle 360 degrees to this portion α, we will

define that if we knew d and we knew α, we can calculate the circumference of the earth.

We got d by timing a camels distance from Syene to Alexandria. Now we just need to find α.

To determine this angle α, let's imagine what it would be like to be standing here in Alexandria.

Well, if I were actually standing there, I can now say the earth is flat

because I don't notice the curvature of the earth on a daily basis

and I can imagine sticking some sort of pole into the ground.

And maybe I know the length of this pole.

Now I know the sun's rays are coming in at a certain angle.

I'm going to draw the rays in red just to make it a little easier to see.

So here comes the sun's rays, you can tell they're being blocked by the pole in some places.

So when the pole blocks the sun's rays, we get a shadowhere's the pole shadow.

And now you can see we're almost there. You can see the right triangle that's emerged.

We have the sun's ray that just barely missed the edge of the pole,

we have the shadow of the pole, and we have the pole itself forming a right triangle.

Now from this drawing, you can actually see that the angle we called alpha.

If I imagine making a perpendicular line here perpendicular to the earth,

this is the angle we called alpha.

There's no problem. That's also equal to this angle α and now we're almost there.

So here's our triangle with three sidesone, two, three, opposite, adjacent to α, and hypotenuse

and we just need to know what this angle is.

Now that we have a right triangle, we just need to consult the trigonometric table

and we should be able to figure out what alpha is and what's the circumference of the earth is.

So we set up the experiment.

We have our vertical bar with its shadow and what Eratosthenes data may have looked like

with the length of the bar would have been something around 1 meter

and the length of the shadow about 0.126 meters which is 12.6 cm.

He, of course, has access to his trigonometric tables and here's a portion of one such table.

Now, can you put yourselves in Eratosthenes' shoes and tell me what is the value of alpha?