Eratosthenes and the Circumference of the Earth

Date: 10/7/95 at 23:8:58
From: Tucker - Joanne
Subject: Ask Dr. Math

Dear Dr. Math,

How did Eratosthenes measure the circumference of the earth?

	Thanks,
	Our Third Grade Class

Date: 10/10/95 at 16:53:35
From: Doctor Andrew
Subject: Re: Ask Dr. Math

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.  
This was about 500 miles.  

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 
points.  If you have any questions about this, please send them to 
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