3. Finding the Circumference of the Earth - ala Eratosthenes

Back to our diagram. If Eratosthenes could figure out (1) the measure of angle ABC and (2) the distance from A to C, using the Pizza problem result he could find the circumference of the earth.

Figure 1

Finding the distance (not to scale) from A (Alexandria) to C (Syene) turned out not to be as formidable a task as circumnavigating the globe and it's the one that Eratosthenes was able to do. But how do he find the angle at the center of the earth? You can't exactly go there and measure it with a protractor. That would be as difficult as digging a tunnel through the earth. But Eratosthenes had a brilliant insight that made this option the one that worked for him. It turns out that the answer lies in studying shadows.

One day at his library Eratosthenes read in a papyrus book that in the frontier outpost of Syene vertical sticks cast no shadow and a reflection of the sun could be seen at the bottom of the well at noon on June 21st. Being the scientist that he was, he wanted to know if the same thing happened exactly at the same time in Alexandria. He was patient and waited until the following June 21 to find out. He discovered that the same thing did not happen; there were shadows in Alexandria. This confirmed his theory that the earth was round.

The key was finding the angle measure of the central angle formed by the extensions to the center of the well at Syene and the vertical stick in Alexandria. Eratosthenes faced two problems (1) how to determine the central angle and (2) finding the distance from Syene to Alexandria. The distance problem was handled empirically by having Behamists (hired mercenaries or 3rd century BC graduate assistants, if you prefer) pace out that distance, but finding the angle had to be done in a clever way from shadow lengths and angles.

Note: A good video to show the students is a segment from the PBS series Cosmos (Volume 1) where Carl Sagan demonstrates the method. You can probably find this video available in your library network.

### Activity: Eratosthenes' Most Amazing Discovery

In this activity your students will discover a relationship involving angles, parallel lines, and a transversal (a line that intersects two parallel lines.) Call up the Javasketchpad sketch. Move Alexandria around to change the angle. (Sorry about all those extra decimals. Its a bug.) What do you notice about angle SA and A? (The two angles are equal.)

Why does that work? Try the paper folding activity.

Eratosthenes concluded that since the sun is so far away the sun's rays are parallel. The line that is formed by the gnomon (equivalent to your meter stick) at Alexandria and the center of the earth, cuts the two parallel lines. The two angles (colored red in the diagram below) are called alternate interior angles and they are equal if formed by this line (a transversal) and two parallel lines.

But what happens if there IS a shadow at Syene and Alexandria? It turns out that this is the case for most of the year. (The reason there was no shadow in Syene on June 21 is because the sun is at its most northernly point directly over the Tropic of Cancer which is 23.5 degrees north of the equator. Syene (today known as Aswan) is located very close to that latitude.

Since for the purposes of this project you will need to find the central angle when there are shadows at both sites, can we determine the measure of the central angle from the two shadows?

• Here there is a shadow at both locations. Collect some data using the Javasketchpad model for different arrangements of cities and central angles. What is the relationship of the sun angles to the central angle? Can you predict what the central angle is if you know the measurements of A and B? (Here is an Activity page that your students can try.) As it turns out, the central angle is equal to the difference between angles A and B. See a proof.

•

Now that we know how to find the central angle, Let's review how to find the circumference.

If you know the measure of the central angle and you can find the distance between town 1 and town 2, you can find the circumference. How many "slices" will fit in this circle if the central angle of each slice is 20.27 degrees? Since there are 360 degrees in the whole circle, then 360 / 20.27 will give you the number of slices. (17.76) If the distance between NJ and PR is 2.5 inches, then the circumference is 2.5 * 17.76 = 44.4 inches.

Activity: Find the circumferences of the following circles

Since there are 360 degrees in a circle, there must be eighteen 20 degree slices (because 360 divided by 20 equals 18.) This means that the circumference must be 18 x 1.8 = 32.4 cm. Try these:

Write a formula for finding the circumference.

** Cosmos #1: The Shores of the Cosmic Ocean
[VHS 697] 1980 Carl Sagan Productions/PBS, 1/2" video, Color 1 cass., 60 min.
Explores the universe from clusters of galaxies to the Milky Way and earth. Discusses early scientific discoveries concerning measurement of the earth's circumference and its spherical nature. Journeys through time from the Big Bang to the present.

(The math lesson ideas end here. Return to the Teachers Guide page.)