Tip:
Highlight text to annotate it
X
In this video, we are going to talk about Eratosthenes
which is a Greek scholar that lived about 2000 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 measurement,
is a simple geometry globe.
So consider the circle shown here
which has points A and B.
And let’s say that we know the distances that A and B
make on the circumference the circle.
So we know the measure of arch 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 any more.
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 around the circle.
Because we know that every circle has
in it a 360°
in one full revolution.
So by knowing data, we know the fraction of the circle
that the arch AB takes up and we can simply extrapolate to find the
circumference.
So, let’s make this one a little clear with the some concrete example.
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°.
And since we know that 90° goes in 360° four times,
arch AB is one-fourth 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
arch AB.
Now let’s just draw this point home further to the other circle.
Again we have points A and B.
And here the angle between points A and B.
Let’s say that we measure and
it turns out to be 36°.
So, since 36° goes into 360° ten times,
we know that ten of these pieces fit in the circle.
And in this case, the circumference is
ten, which is the number of pieces,
time 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 writing down specifically, the number of pieces is
360°
divided by data,
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 arch 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 your
Eratosthenes method.
So, now let’s apply this.
Now we have another circle. But this time, we will specifically identify the circle as
the Earth.
And point A becomes the city Alexandria.
This, a city in Egypt
and this is where Eratosthenes lived.
Point B now becomes the city of Syene.
So in Eratosthenes’ 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 factors, 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 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 a summer solstice
you can see the sun’s rays light up at the bottom of the well.
And if you think about this, what this means is that since the well is such a deep
thing,
this means that the sun rays must’ve been coming into the Earth,
parallel to the well.
So we draw these rays,
since sun is 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 ray that
I’ve also shown, this angle we’ve called data 2
is equal to the angle that we are trying to find, data 1.
This is something, it’s a fact that you might have been exposed to in your geometry classes,
it’s called property of corresponding angles,
but we’re gonna say it’s pretty easy to prove it yourself.
So, since we know this,
now we can transform the hard problem of measuring data 1
to the relatively easier problem of measuring data 2.
So how do we measure data 2?
It’s pretty straightforward. So let’s zoom in on this region.
We have hit 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,
data 2. All we have to do is to look at the shadow created off of the stick.
And this is exactly what Eratosthenes did.
And he measured that the angle, data 2 is equal to
7.2°.
So, now we have this information, the angle,
the data, which is called the data now,
is equal to 7.2°.
And the measure of the arch AB or Alexandria and Syene
is 500 miles.
So, with this information now, all we have to do is feed this into the formula
we got earlier, let’s just call it C equals
360° divided by data,
times, AB.
So, this is very simple to do.
Plug in 7.2° for data. Plug in 500 miles
for AB.
And in the end we find 50 times 500 miles,
or 25,000 miles.
So this is our estimate or Eratosthenes’ 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, it’s around 24,900 miles.
So we are only about a 100 miles off with this seemingly primitive method,
which is about half a percent off.
So this is very impressive thing for someone who lived so long ago without
the access to these tools that we now use.