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### Eratosthenes and the Circumference of the Earth

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Date: 10/7/95 at 23:8:58
From: Tucker - Joanne

Dear Dr. Math,

How did Eratosthenes measure the circumference of the earth?

Thanks,
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Date: 10/10/95 at 16:53:35
From: Doctor Andrew

Well, according to the Encyclopedia Americana and the Encyclopedia
Britannica, Eratosthenes observed that at noon on the summer
solstice (the longest day of the year) the sun was directly
overhead in the city of Syene in Egypt (it is called Aswan now).
I've heard elsewhere that he knew this because at that time, no
shadow was cast in a well. A well isn't necessary to observe this
though, since any container with parallel walls such as a box or a
tube will not have a shadow when light comes from directly above
it.  Try it out yourself.

He then assumed that the sun was so far off that its rays hit the
earth in parallel.  If you imagine all the lines from the surface
of one ball to another, you can see that as the balls get further
apart, all the lines become nearly parallel.  Parallel means that
the lines all go in exactly the same direction.  He also assumed
that the earth was shaped like a ball.

He also knew that Syene was on the same meridian as the city of
Alexandria. The earth is a ball that spins around a line called
its axis. A meridian, (also called a line of longitude) is a line
on the surface of the earth from one end of this axis to the
other. I'm not sure how he knew these two cities were on the same
meridian; maybe he knew that the sun set at the same time when it
was directly between the two cities. If you follow a meridian all
the way around the earth you get a circle, like the equator.

Finally, Eratosthenes knew that the distance between Alexandria
and Syene was 5000 stadia, a Greek unit for measuring length.

So, on the summer solstice, at noon, in Alexandria, Eratosthenes
measured the angle of the sun's rays.  You could do this by
finding the angle at which a shape casts the least shadow.
Suppose you had a globe that had a metal band around it that could
rotate around the globe but could also be completely horizontal
(globes usually have bands around them that are vertical).  You
may have one like this in your classroom.  If you take the globe
out and then rotate this band until its shadow is only a line, it
will be parallel to (in the same direction as) the rays of the
sun.  Think about what fraction of a whole circle you had to
rotate the band.  Well, Eratosthenes probably had a device similar
to this which he had to rotate 1/50 of a whole circle to get it to
line up with the sun's rays.

Using a little geometry (that is a little tough for 3rd grade) he
then knew that 5000 stadia was 1/50 of the circumference of the
earth.  This means that he needed to use 50 of these lengths to
surround the earth.

So he multiplied 5000 by 50 to get 250,000 stadia.  Then he added
2000 more to make up for what he thought were bad measurements.
So he calculated the circumference of the earth to be 252,000
stadia.  We know that the distance between Alexandria and Syene is
about 500 miles, so using his fraction 1/50, we can get the
circumference of the earth to be about 500 * 50 = 25000 miles,
which is about right. Since historians aren't sure how long one
stadia is, we haven't been able to figure out how close
Eratosthenes was to the correct answer, but we do know that the
way he tried to solve them problem was correct.  In those days
they couldn't easily make very accurate measurements of the
distance between two places, so this could cause a lot of error.

I hope this is all clear, but there are probably some messy
us.  If you want to know how Eratosthenes used Geometry to show
that the 1/50 of a circle on the angle measuring device means that
the distance between the two cities was 1/50 of the circumference
of the earth, I'd be glad to try to explain it.

-Doctor Andrew,  The Geometry Forum

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Associated Topics:
Elementary Circles
Elementary Geometry
Elementary Math History/Biography

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