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