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### 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/
```
Associated Topics:
Middle School Conic Sections/Circles
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School History/Biography
Middle School Measurement

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