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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]