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# Ancient Techniques of Determining Earth's Size and Shape

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Author: Janice VanCleave

In ancient times many natural phenomena were explained by weaving myths about what was observed. As time passed, more practical applications were based on observations, such as using stars as points of reference when traveling. The Greeks were noted for applications of geometric measurement that were amazingly accurate in describing the size and motion of the Earth and other planets.

In this project, you will calculate the circumference and radius of a circle using the geometric method Eratosthenes used to determine the circumference and radius of the Earth. You will learn how to calculate the angle of a shadow. Following Eratosthenes' example of using the difference in the angle of shadows cast in different cities at the same hour, you will determine the circumference of the Earth for yourself. You will learn how a lunar eclipse was used to verify the shape of the Earth.

### Getting Started

Purpose: To learn Eratosthenes' geometric method of determining the arc between two cities on the same meridian.

### Materials

• Scissors
• Sheet of typing paper
• Pen
• Metric ruler
• Protractor

### Procedure

1. Lay the roll of tape flat in the center of the paper and trace around it with the pen.
2. Find the center of the circle drawn on the paper by folding the circle in half twice: first fold the circle from top to bottom, then fold again from side to side.
3. Unfold the paper and mark a point in the center where the fold lines cross. Label this point A.
4. Lay the ruler across the circle with its bottom edge on the horizontal fold line.
5. Mark two points on the circumference of the circle where the top and bottom edges of the ruler touch the one side of the circle. Label the points B and C, as shown in Figure 8.1.
6. Use the ruler to draw a line from point A to each of the points B and C. Extend the lines 5 cm or more outside the perimeter of the circle to points D and F (see Figure 8.1).
7. From points B and D, draw lines perpendicular to the circle and parallel to line FC. Mark point E as shown.
8. Use the protractor to measure the angles between angle CAB and EBD (see Figure 8.1).

### Results

The degrees of an arc are determined.

### Why?

Eratosthenes (276-194 B.C.), a librarian at the museum in Alexandria, Egypt, used a geometric method similar to the one in this experiment to determine the degrees of arc (part of a circle) between two cities, Syene (Aswan) and Alexandria. He believed that the Earth is a sphere. His method involved using the difference in the angle of shadows cast at the same hour in the cities. Eratosthenes learned that at noon on the summer solstice (June 21), the Sun's reflection could be seen in the water at the bottom of a well in Syene (Aswan). This meant that the Sun was exactly overhead at that time and no shadows were cast. Thus, the Sun's rays were perpendicular to the well and in line with the radius of the Earth, represented by line CA in this experiment. He observed that at the same time in Alexandria, a tall pillar cast a shadow. Eratosthenes knew that the pillar was perpendicular to the Earth's surface and thus in line with the radius of the Earth. Since sunlight comes from such a great distance, sun rays are parallel to each other when they reach the Earth. Because the Earth's surface is curved, there is an angle between the pillar and the parallel Sun's rays. This angle is represented by the shadow angle in this experiment, which is equal to angle EBD and angle CAB at the center of the circle. With this information, he determined the angle of the arc between Syene and Alexandria using the angle of the pillar's shadow. In this experiment, angle CAB, angle EBD, and the shadow's angle are 31°, but the angle measured by Eratosthenes was only 7°. Thus, the arc between the two cities was determined to be 7°.

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