For centuries educated people believed that the earth was a sphere.  The circumference of the sphere we call the earth was first measured by Eratosthenes (c. 276-194 BCE), the third Librarian of the famous Library at Alexandria, Egypt.  A scholar in his own right but not the brightest (he was called Beta because he was at the second level), he wrote a famous book which explains the mathematics underlying the philosophy of Plato.  What Eratosthenes actually measured was the polar circumference of the earth.  While there are at  least two stories about how he did this, the one described here provides secondary students with an activity whereby they, too, can do the mathematics that comes very close to the measure of the actual polar circumference.  Among the places where this activity can be used are in an arithmetic class when the circumference of circles is discussed and in geometry classes where angles and parallel lines are the topic.  For the activity students will need pencils and a handout showing a circle marked as in Figure 1.

Figure 1

            The advantages for the students are many.  After having located two significant places in Egypt (Alexandria and Syene, now called Aswan), they draw representative components of the activity;  that is, they construct a mathematical model of an historical problem.  They will apply a significant theorem (interior angles formed by a transversal crossing two parallel lines are equal), a crucial relationship (two ratios make a proportion), and an important formula (circumference equals pi times the diameter).  Hopefully, they will feel confident enough to explain the lesson to their parents.

            Together with a transparency or wall map of Egypt to enable the students to see where the two cities of Alexandria and Syene are located, the teacher will need a transparency of Figure 1 (or s/he could draw the circle and its parts on the chalk board) for completing the picture of the problem (Figure 2) as s/he tells the story.   Further, as the teacher completes the picture (Figure 2) on the chalk board, the students copy the various lines onto their handout.

Figure 2

            The teacher begins by locating the two cities on the map, noting that Syene (S) is now called Aswan where the Egyptians built a dam to help control the flooding of the Nile River.  The letter A is for Alexandria.  Both cities lie practically on the same meridian about  5,040 stadia apart.  A stadium is an ancient Greek unit of measurement equal to about 516.7 feet or 157.5 meters.  At noon the midsummer sun is directly over Syene.  Hence, a vertical object casts no shadow, nor is there any shadow in the bottom of a well.  However, at Alexandria a pillar does cast a shadow.  And so the angular distance between Alexandria and Syene can be measured.  The angular distance is determined by imagining a point at the center of the earth, C in Figures 1 and 2, to which line-segments from A and S are drawn;  thus a central angle is formed.  The number of degrees in this angle at the center of the earth can be found; and this is the angular distance on the surface (or circumference) between the two points A and S.