Search Loci: Convergence:
What is this frog and mouse battle among the mathematicians?
In H. Eves Mathematical Circles Squared Boston: Prindle, Weber and Schmidt, 1972.
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Measuring the Globe: An Historical Activity
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 very 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 or algebra 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.
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 an image 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 an image of Figure 1 on which s/he can complete the picture of the problem (Figure 2) as s/he tells the story. Further, as the teacher completes the picture (Figure 2), the students should complete the picture as well on their handouts.
The teacher begins by locating the two cities on the map, noting that Syene (S) is now called Aswan and is the site of a dam built by the Egyptians to help control the flooding of the Nile River. The letter A is for the city of 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 at 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.
We assume that the rays of the sun strike the earth parallel to each other, the broken lines in Figure 2. The ray that goes through the tip of the pillar forms an angle with the pillar itself. At the same time a parallel ray is hitting the bottom of the well in Syene and, as it were, continuing to the center of the earth. The well-ray makes an angle with the (solid) line-segment that goes from the center of the earth to the base of the pillar. These two angles, at C and at the tip of the pillar, are equal because of the theorem: If a transversal (the solid line-segment) cuts two parallel lines (the broken lines), the alternate interior angles are equal. So, by knowing the measure of the angle at the tip of the pillar, you know the central angle at C; hence, you know the number of degrees between the well at Syene and the pillar in Alexandria. How did Eratosthenes measure the angle at the tip of the pillar? No one really knows; but he did and he did it remarkably well: 7 1/5˚. Today, we could measure the length of the shadow of the pillar, divide that number by the measure of the height of the pillar, and thereby find the tangent of the tip-angle. We could then find the tip-angle using a calculator or a table of tangents. Eratosthenes had neither a calculator nor a table of tangent ratios because they had not yet been invented.
Nonetheless he did find that the tip-angle was 7 1/5˚, making the angle at the center of the earth 7 1/5˚ and the arc at the surface 7 1/5˚ or 1/50 of the polar circumference (P) of the earth. There is enough information to create two ratios: the first compares the degrees from A to S and the total number of degrees in the circumference, or 7 1/5 : 360. The second compares the Alexandria–Syene distance and P, or 5040 : P. Since the two ratios are talking about the same thing, the measure of an arc of the circle around the earth, they can be set equal to one another as a proportion:
7 1/5 : 360 = 5040 : P .
This solves the problem. The computation of P is left to the students to complete on their handouts.
The problem becomes even more enriching as the teacher explains to the students how necessary it is to simplify facts in order to solve problems. For instance, the earth is not a perfect sphere but nearly so, nor are the sun's rays exactly parallel but nearly so. To be absolutely exact would be to make many problems impossible to solve. As an additional activity, the students can use this information to determine the diameter and radius of the earth.
If the teacher wishes to perform an analogous experiment with the class, she should make arrangements with a school several hundred miles away, either directly north or directly south of her school. At astronomical noon on a given day, both schools should measure the altitude of the sun. By using a diagram similar to Figure 2 above, the students should be able to convince themselves that the difference in the altitude measurements is equal to the difference in degrees along a meridian between the two schools. The next step is to measure the actual distance (in miles or kilometers) between the two schools. In Eratosthenes’ time, this measurement was done by pacing off the distance. Today, that might be more difficult, especially in a limited time. So one can “cheat” a bit and find the distance by consulting a map. In any case, with the actual distance, say d, known, as well as the difference in degrees, say θ, students can set up a proportion to determine the polar circumference P of the earth:
d : P = θ : 360
Although this method of determining P is theoretically correct, there are a couple of real-life problems to solve. First, one has to determine astronomical noon. This is defined to be the time when the sun is at its maximum altitude for the day, or, equivalently, when the shadow of a particular object is at its shortest. Thus, one can look up this time (and it probably will not be 12:00 on your clock) in an astronomical almanac, or on an appropriate website, or one can trace the moving shadow of a convenient pole and determine when it is shortest. The second problem is then to measure the sun’s altitude. There are instruments which one can use to sight the sun and determine its altitude (but under no circumstances should one look directly at the sun). Alternatively, you can use trigonometry to determine the sun’s angle, given the length of your pole and the length of its shadow at noon. Of course, as students will discover, there is error in either of these determinations. So students should discuss the possible errors and figure out how to minimize them. They might also want to think about how Eratosthenes was able to get such an exact figure for both the distance between Alexandria and Syene and his angle measurement. A class discussion of the application of theoretical mathematics to the real world may well be of interest to wrap up this lesson.