Measuring the Circumference of the Earth

Date: 08/16/99 at 20:46:44
From: Maria
Subject: Geometry (World Geography)

Hi,

I have to prove the earth is round using a mathematical formula easy 
enough for any family member to use. My proof must contain a diagram, 
a mathematical formula, a solution to that formula, and the answer. I 
have done the other half of the research, which involved finding the 
first people to prove the earth was round (how they did so and when). 
I really need your help.

Thank you so much,
Maria

Date: 08/17/99 at 12:21:20
From: Doctor Peterson
Subject: Re: Geometry (World Geography)

Hi, Maria.

Technically, you can't prove the earth is round using only math; you 
need some observations of the real world. I suspect that you are 
expected to report about how Eratosthenes determined the size of the 
earth in ancient Greek times. For that information, you could search 
either in Dr. Math or on the Web as a whole for "Eratosthenes" and 
"Earth." Basically, he knew the angle of the sun at the same time in 
two places a known distance apart, and from that determined the radius 
of the earth using a simple formula.

But that really isn't an answer to the question as stated, and perhaps 
pointing this out might stir up some good discussions in your class. 
The fact is, Eratosthenes already KNEW that the earth was round; his 
calculations didn't prove it, and couldn't. Here's how you can show 
that this is true: find the data he used, then assume that the earth 
is flat and the sun is some distance above the surface at noon. See if 
you can calculate how high the sun is. You'll find that you can get an 
answer that works just as well as the answer he gave for the size of 
the earth. (There's an interesting relation between the two answers, 
too.) So if I believed that the earth was flat, I could give as good 
an explanation of his experiment as he could assuming that the earth 
is round. Nothing has been proved.

In reality, the ancient Greeks and others knew the earth was round 
based on more general knowledge: that the earth's shadow on the moon 
during a lunar eclipse is round; that ships disappear over the 
horizon; and so on. But none of these involve a formula. Finding 
Eratosthenes' method is undoubtedly the intention of your assignment. 
You can find the details in our archives or elsewhere.

Since the explanation in our archives doesn't give a picture or the 
formula, here's a drawing of his setup. The sun is directly overhead 
in one place and at an angle in another place:

                             |
                             |             /
                             |            /|
                             |           /A|
                   no shadow |    d     /  |
                        ***********    /   |
                  ******     |     ******  |shadow
               ***           |       /   ***
            ***              |      /       ***
           *                 |     /           *
         **                  |    /             **
        *                    |   /                *
       *                     |  /                  *
      *                      |A/                    *
     *                       |/                      *
     *                       *                       *
     *                                               *
     *                                               *
     *                                               *
      *                                             *
       *                                           *
        *                                         *
         **                                     **
           *                                   *
            ***                             ***
               ***                       ***
                  ******           ******
                        ***********

If you know the distance d to the place where the sun is directly 
overhead, and the angle A that the sun makes here, then we can say 
that

      d     A
     --- = ---
      C    360

where C is the circumference of the earth. Solve that for C and you're 
done.

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