How Did Eratosthenes Measure the Circumference of the Earth?

Date: Wed, 26 Jun 1996 12:25:35 -0400 (EDT) 
From: Ed and Roxann
Subject: Eratosthenes' method to find Earth's circumference 

How did Eratosthenes measure the circumference of the earth? 

As I understand the process, he measured the lengths of shadows of 
sticks at different locations on the same day and time of the year, so 
he had two right triangles, but they weren't similar. So what do I 
do?

   - Ed and Roxann

Date: Wed, 26 Jun 1996 13:51:03 -0400 (EDT) 
From: Dr. Ethan
Subject: Re: Eratosthenes' method to find Earth's circumference 

This will be a little hard to explain without a picture but I will do 
my best.

Eratosthenes made a few assumptions.  One was that the earth was a 
sphere and the other was that the sun's rays were parallel.  Neither 
assumption is true, but both were close enough.  We will make the 
same assumptions 

If the earth were flat and the sun's rays were parallel, if you put 
two sticks straight up a half mile away from each other then their 
shadow would make the same angle.  But if the earth were curved, 
then the angle made by the shadows would be different.  This 
difference can be used to figure out the circumference of the earth.

Let's try an example.  Let's say you are Eratosthenes in Athens and 
you have a stick sticking straight into the ground, and at high noon 
the sun is straight overhead so it leaves no shadow.  But your friend 
is in Rome which is (I'm guessing) 400 miles away, and there at the 
same time the sun makes a shadow at a 30 degree angle.  That means 
that in 400 miles what appears to be straight up and down is 30 
degrees different from what it does in Athens. 

This means that you have moved 30 degrees around the Sphere (Earth)
in those 400 miles.  Well, since there are 360 degrees in a circle
and 30/360 = 1/12, then 400 must be 1/12th of the total circumference
of the Earth.  So the circumference of the Earth is 12 * 400 or 
4800 miles. 

Now take note: these numbers are made up and do not reflect 
Eratosthenes' calculations or the actual circumference of the Earth. 

Hope that helps.

-Doctor Ethan, The Math Forum
  http://mathforum.org/dr.math/   

Date: Wed, 26 Jun 1996 13:34:52 -0600
From: Ed and Roxann
Subject: Re: Eratosthenes' method to find Earth's circumference 

Thanks, it's so easy now. I tend to get myopic when I start down 
the wrong path.

Date: 06/15/2005 at 15:20:28
From: Greg
Subject: Eratosthenes, circumference of earth and margin of error

I have just read your version of the Eratosthenes story about 
finding the circumference of the earth and have seen it twice before. 

If I understand the story appropriately Eratosthenes was in Syene or 
Aswan and knew when the sun cast effectively no shadow, but how could 
he figure out the angle of the shadow at that same moment in 
Alexandria, 500 miles away?  Also, the 500 miles is close to the 
correct distance but not as the crow flies, or not exactly as the arc 
of the earth.  The timing, distance and rough measurements all seem to 
add up to an inaccurate but ingenious approach.  Could you explain how 
he was so accurate given the margin of error of all the calculations?

My thoughts are the timing of the shadow needs to be fairly accurate 
because the sun is in constant motion so the angle would be constantly
changing.  The ground and route taken from Syene to Alexandria is not 
exact either, what if the route was measured by circumventing natural 
obstacles or by measuring both up and down a hill?  He was apparently 
within 4 percent, but that seems more luck than anything, even given 
Fermi's reasonableness estimations.

Date: 06/15/2005 at 20:34:29
From: Doctor Rick
Subject: Re: Eratosthenes, circumference of earth and margin of error

Hi, Greg.

Here is one site that explains the situation in some historical 
detail:

  Measuring the Solar System (Michael Fowler, UVa)
  http://galileoandeinstein.physics.virginia.edu/lectures/gkastr1.html

I can't answer all your questions, but I can at least put one to 
rest.  There is no significant issue with the timing of the sightings. 
If the sun is directly overhead, it must be noon (well, what 
navigators call "local apparent noon") when the sun is highest.  The 
sun can't be any higher than straight overhead.  It was known that on 
a certain day (the summer solstice) the sun passed directly overhead 
at Syene.  All that was necessary in regard to timing was to measure 
the elevation of the sun at its highest point on that day.

In celestial navigation, this is called a "noon sight" and is the 
easiest way to use a sextant to determine something about your 
position.  Specifically, it tells you your latitude.  All you need is
a sextant and a table that gives the sun's elevation on each day of 
the year (or at closer intervals if you need more precision).  You 
take a sight of the sun and continue adjusting the sextant until the 
sun starts going down.  The highest value you found is the sun's 
elevation at local apparent noon.

The important thing about a noon sight is that you don't need a 
precise timepiece to do it.  For other sights you do need to know 
exactly when you took the sight, because the sun moves east to west 
a lot faster than it moves north and south, due to the earth's 
rotation.  This is the reason that old maps have much more accuracy 
in their latitudes than in their longitudes: accurate chronometers had 
not been developed.  Dava Sobel's book "Longitude" tells the 
fascinating story of how this situation was remedied at last.

Back to Eratosthenes, a factor that you didn't mention is the 
accuracy of the observation that Syene is due south of Alexandria. 
That's a second-order error, though (the error in the result varies 
as the square of the error in east-west position); I agree that the 
error in measuring the distance if the roads weren't straight is 
probably the major source of error.  But I don't know how great his 
errors were, or even whether we can know.  The site I linked above 
says that the length of Eratosthenes' unit of distance, the stade, is 
not known for sure, so we can't be sure how accurate his result was.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/