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# Circumference of Earth

## Video transcript

[MUSIC PLAYING] In this video, we're going to talk about Eratosthenes, who was a Greek scholar that lived about 2,000 years ago. Eratosthenes found a way, using none of the modern tools that we have, to measure the circumference of the Earth. And in this video, we're going to see how he did this. So the heart of Eratosthenes's measurement is a simple geometry problem. So consider the circle shown here, which has points A and B. And let's say that we know the distance that A and B make on the circumference of the circle. So we know the measure of arc AB. Now, the question is, with this knowledge, can we determine the circumference of our circle? And I'm sure you all are thinking that the answer is obviously no because A and B are just two random points on the circle. So just knowing their distance doesn't help us very much. What we need is some information that makes A and B not just random points on the circle anymore. We need to know the angle that A and B make with the center of the circle. Once we know this, we know how far around A and B go on the circle because we know that every circle has in it 360 degrees in one full revolution. So by knowing theta, we know the fraction of the circle that the arc AB takes up, and we can simply extrapolate to find the circumference. So let's make this one a little clearer with some concrete examples. So let's look at the circle on the right now. And again, we have points A and B. And here you can clearly see that the angle that A and B make with the center of the circle is 90 degrees. And since we know that 90 goes into 360 four times, arc AB is 1/4 around the circle, meaning that the piece shown here, the shaded piece, will fit into the circle four times. So in this case, the circumference of the circle is four times the length of arc AB. Now let's just drive the point home further with this other circle. Again, we have points A and B. And here the angle between points A and B, let's say we measure, and it turns out to be 36 degrees. So since 36 goes into 360 ten times, we know that 10 of these pieces will fit into the circle. And in this case, the circumference is 10, which is the number of pieces, times the length of one piece, AB. Now, in general, the circumference of the circle is given by the number of pieces we have times the length of one piece. And just right inside explicitly, the number of pieces is 360 degrees divided by theta, the angle that A and B make with the center of the circle. And the length of one piece is simply AB. So now, if we just know these two things, the arc AB and the angle A and B make with the center of the circle, we can determine the circumference. And this is really the heart of Eratosthenes's method. So now let's apply this. Now we have another circle, but this time we'll explicitly identify the circle as the Earth, and point A becomes a city, Alexandria. This is a city in Egypt, and this is where Eratosthenes lived. Point B now becomes the city of Syene. So in Eratosthenes's day, it was known that the distance between Alexandria and Syene was about 500 miles. Of course, back then the units weren't miles. But we know the conversion factor, so we don't have to worry about the old system of units. The distance between Alexandria and Syene is 500 miles. And now, looking back at our previous problem, we see that all we have to do to figure out the circumference of the Earth given this information is to figure out the angle that Alexandria and Syene make with the center of the Earth. And the really brilliant thing about Eratosthenes's method is that he found a nice way to measure this angle. So how did he do this? It was known that in Syene there was a well, a long, deep well such that at noon on the summer solstice you could see the sun's rays light up in the bottom of the well. And if you think about this, what this means is that since this well is such a deep thing, this means that the sun's rays must have been coming into the Earth parallel to the well. So we draw these rays, and we see the sun is very, very far away from the Earth. And because the sun is so far away, we can treat the rays coming in from the sun at different points as parallel. So here we've drawn the ray that comes in at Alexandria and the ray at Syene. Now, it's an interesting property of parallel lines that if we have these two parallel lines shown here, the two rays, and we have the radii also shown, this angle that we've called theta 2 is equal to the angle that we're trying to find, theta 1. This is a fact that you might have been exposed to in your geometry classes. It's called the property of corresponding angles. But regardless, it's pretty easy to prove to yourself. So since we know this, now we can transform the hard problem of measuring theta 1 to the relatively easier problem of measuring theta 2. So how do we measure theta 2? It's pretty straightforward. So let's zoom in on this region. We have the surface of the Earth as one of the important lines. And then the radius gets translated into a vertical stick. And we also have the sun with one of its rays. So this ray casts a shadow on the ground. And by knowing the length of the shadow and the height of our stick, we can construct this triangle and simply measure the angle theta 2. All we have to do is look at the shadow created off of a stick. And this is exactly what Eratosthenes did. And he measured that the angle theta 2 was equal to 7.2 degrees. So now we have this information. The angle theta, which is called theta now, is equal to 7.2 degrees. And the measure of the arc AB, or Alexandria to Syene, is 500 miles. So with this information, now all we have to do is feed this into the formula that we got earlier. Let's just recall it-- C equals 360 degrees divided by theta times AB. And so this is very simple to do. We plug in 7.2 degrees for theta. We plug in 500 miles for AB. And in the end, we find that C is equal to 50 times 500 miles, or 25,000 miles. So this is our estimate, or Eratosthenes's estimate, for the circumference of the Earth. Let's call that C Eratosthenes. And so see how simple it was for us to get this. But notice that, according to our modern measurements, the average circumference of the Earth, because it's not a perfect sphere, is around 24,900 miles. So we were only about 100 miles off with this seemingly primitive method, which is about half a percent off. So this is a very impressive thing for someone who lived so long ago without the access to these tools that we now use. [MUSIC PLAYING]