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# Ancient Techniques of Determining Earth's Size and Shape (page 2)

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Author: Janice VanCleave

### Try New Approaches

1.
d1/d2 = A1/A2

where

1. The degrees of arc and the arc length (length of a portion of the circumference of a circle) can be used to determine the circumference of a circle. Determine the circumference of the roll of tape. First measure the arc length between points B and C in the drawing by laying the roll of tape on the circle and marking the positions of points B and C on the tape. Unroll enough of the tape to measure the distance between the two marks. Record this distance as d2. In the example, this distance is 2.9 cm. Then, use the following equation and example to calculate the outer circumference of the roll of tape:
• dl = circumference of the circle
• d2 = distance between points Band C on the circle (= 2.9 cm in this example)
• A1 = angle of a circle (= 360°)
• A2 = angle of the portion of the circumference between points B and C (= 31° in this example)
2. Determine the Earth's circumference as calculated by Eratosthenes using the following information and the equation from the previous experiment.
• dl = circumference of the Earth
• d2 = distance from Syene to Alexandria (= 500 miles ([800 km])
• Al = angle of a circle (= 360°)
• A2 = angle of the portion of the arc between Syene and Alexandria (7°)
2.
d1 = 2πr

where

1. Eratosthenes knew that the circumference of a circle is about 3.14 times as great as its diameter. The Greeks called this number pi, which is written π. He used pi and the fact that the diameter of a circle is twice its radius to calculate the Earth's radius. Use the following equation and example and the circumference from the previous experiment to determine the Earth's radius as calculated by Eratosthenes:
• d1= circumference of the circle (= 33.7 cm in the original experiment)
• π = 3.14
• r = radius of the circle
2. Determine the Earth's radius as calculated by Eratosthenes using the following information and the equation from the previous experiment.
• d1= circumference of the Earth (calculated in part 1a)
• r = radius of the Earth
• π = 3.14
3. Eratosthenes calculated the Earth's polar circumference, the distance around the Earth from Pole to Pole. This circumference is now known to be about 24,951 miles (39,922 km), and the radius is about 3,973 miles (6,357 km). Use these figures and the information in Chapter 5, "Night Light: The Structure and Movement of the Earth's Moon," to calculate the percentage of error for Eratosthenes' calculations for the Earth's circumference and radius.

1. Demonstrate how the angle of a shadow is calculated. Tape a ruler to the side of a wooden block so that the ruler stands vertical. Set the block outdoors on level ground and use a carpenter's level to ensure that the block is level. Tape a string to the top of the ruler, and without moving the ruler, have a helper stretch the string between the top of the ruler and the end of the ruler's shadow. Use a protractor to measure the angle between the ruler and the string, as shown in Figure 8.2.
2. Use Eratosthenes' method to measure the circumference of the Earth yourself, using a helper who lives several hundred miles (km) due north or south of you. First, use an atlas to determine the longitude and latitude of your town. Then find a town on the same longitude about 50 to 100 of latitude north or south. Use the scale on the map to determine the exact distance in miles. If you do not know anyone who lives in that town, with adult assistance you might be able to find someone via the World Wide Web. Design a way to measure the angle of the shadow, or use the method from the previous experiment. You must measure the angle of the shadow at the same hour. Try to do it on the same day or within the same week. Determine the absolute difference of the angle of the two shadows by subtracting the smaller angle from the larger angle. Use the equation from the original experiment to calculate the circumference of the Earth.

Take measurements two to three different times during the day and average the circumferences from your results. Repeat the procedure with a helper in another town in the opposite direction. Compare your measurements to the modem measurement of the Earth's circumference, calculating your percentage of error. Make and display diagrams showing the angles of the shadow measured by Eratosthenes and those measured by you and your helpers.

### Get the Facts

1. Eratosthenes' calculations were replaced by less accurate measurements promoted by Ptolemy (second century A.D.), an Alexandrian scholar. Ptolemy calculated the circumference to be about 18,000 miles (28,800 km). Because of this inaccurate measurement, Christopher Columbus (1451-1506) sailed west thinking the Earth was much smaller than it really is. What was the system of measurement promoted by Ptolemy? For information about early measurements of the Earth's circumference, see John Farndon, How the Earth Works (New York: Reader's Digest Association, 1992), p. 22.
2. Some early astronomers thought the Earth to be a flat square under a pyramid-shaped sky, and others believed it to be a plate resting on the backs of four elephants standing on a giant floating turtle. Find out more about the early views of the shape of the Earth and its place in the universe. Make and display diagrams of some of the early views. How was a lunar eclipse used to prove the Earth is round? For information, see How the Earth Works, pp. 12-13,24.
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