Eratosthenes: Estimating the Circumference of the Earth. Subject: Mathematics Topic: Geometry Grade Level: 8-12 Time: min Pre Show Math Activity

Transcription

1 Eratosthenes: Estimating the Circumference of the Earth. Subject: Mathematics Topic: Geometry Grade Level: 8-12 Time: min Pre Show Math Activity

2 Introduction: In Show Math, students will learn how in 200 BC, the Greek mathematician Eratosthenes calculated the circumference of the Earth using simple geometry and a few measurements. His estimate of the circumference was remarkably accurate to within 2% of the real value. In this activity, students will follow in Eratosthenes footsteps by finding the circumference of the Earth themselves using sticks, shadows, and abstraction. It is an activity of discovery, where the students will piece together the geometry for themselves, using Eratosthenes s assumptions and the few pieces of information he had available. Learning Objectives: By the end of this activity students should: Understand how abstraction can help solve math and real life problems Explore the problems encountered when measuring the Earth s circumference Examine the ways that errors will occur by measuring circumference in this way Skills: Students should develop: Team work skills Communication adn collaboration skills Abstract thinking skills Material & Resources: Chart Paper WNCP Curriculum Links: Mathematics 9 Apprenticeship and Workplace Mathematics 10 Foundations of Mathematics 10 Solve problems and justify the solution strategy using circle properties. [C, CN, PS, R, T, V] Analyze puzzles and games that involve spatial reasoning, using problem-solving strategies. [C, CN, PS, R] Solve problems that involve parallel, perpendicular and transversal lines, and pairs of angles formed between them. [C, CN, PS, V] Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles. [C, CN, PS, R, T, V]

3 Foundations of Mathematics 11 Apprenticeship and Workplace Mathematics 12 Solve problems that involve the properties of angles and triangles. [CN, PS, V] Demonstrate an understanding of the limitations of measuring instruments and solve problems. [C, PS, R, T, V] Background: In ancient times it was widely believed that the world was flat with the heavens a physical dome spanning over it. However, there were some early arguments for a spherical Earth. For example, it was observed that during a lunar eclipse the Earth cast a circular shadow of the Earth on the moon during a lunar eclipse. It was also noted that the star Polaris is seen lower in the sky as one travels South. Pythagoras observed that ships masts appear before their hull when they appear over the horizon. Since these phenomena are observed everywhere, it suggests that the earth is curved everywhere. The logical result is that the earth must be spherical. Archimedes and Plato both believing the Earth to be a sphere, tried to estimate the circumference of the Earth. However, the first to give an accurate calculation of the circumference of the Earth was Eratosthenes in about 240 BC. Eratosthenes knew that in Cyene, Egypt (now known as Aswan) on the summer solstice, when the sun its zenith, sticks and pillars cast no shadow. However, in his home town of Alexandria, he saw that a stick planted in the ground cast a shadow measuring 7.2 degrees. Since the sun is very far away, when its rays hit the Earth they are nearly parallel. Therefore, if the Earth were flat and if a stick in Cyene cast no shadow at noon, then a stick in Alexandria would also cast no shadow. Because the stick in Alexandria does cast a shadow, he could use this information to estimate the earth s circumference Imagine we extend the sticks until they meet at the centre of the Earth. We can see that since the sun s rays are parallel and that alternate angles are equal, we can see that Alexandria and Cyene are separated by an angle of 7.2 degrees.

4 Eratosthenes paid someone to pace out the distance between Cyene and Alexandria which he found to be 5,000 stadia - approximately 800km. Now using proportion he found the circumference using the following equation: Eratosthenes did some rounding to 700 stadia per degree and implied a circumference of 252,000 stadia. If Eratosthenes used the Egyptian stade of about metres his measurement turns out to be 39,690 km, an error of less than 2%. The circumference of the Earth between the North and South Pole is 40,008 km. There are limitations in the accuracy of Eratosthenes calculation. The accuracy of his measurement is reduced by the fact that Cyene is not directly south of Alexandria and that the sun rays are not perfectly parallel when they hit the Earth. However, the greatest limitation of Eratosthenes method is that his distance measurement was not reliable because the roads connecting the two cities were not a straight line path (great arc, actually) between them. Given the margin of error in these aspects of his calculation, the accuracy of Eratosthenes circumference estimate is surprising. Resources: A description of Eratosthenes experiment Details the history of the beliefs about the Earth s shape and size A video that explains Eratosthenes methods Activity Instructions: 1. To class say: Today we are going to go back in time to Ancient Egypt. We need to find the circumference of the Earth without any modern technology. Any ideas how we could do this? 2. Tell class that in Ancient Times, Eratosthenes used a few observations and geometricalreasoning to deter mine the circumference. Put yourself in the Eratosthenes shoes. Here s what you know and what you assume: Display the overhead (Figure 1) of information available to Eratosthenes. You may also want to print off some copies of Figure 1 for the groups. 3. Divide class in groups of 4-5 students. Given them each a piece of chart paper and have each group draw a labelled diagram of the sticks in the earth and the shadow. 4. Allow each group to share their diagrams with the class. If necessary, provide hints using the overhead of the geometry (Figure 2), in case the student diagrams aren t quite right.

5 5. Tell the students to go back into their groups and tell them to use the diagram to find the circumference of the Earth. Circulate through out the class and help groups as they need it. 6. Allow each group to share their calculations. If necessary solve the problem on the board. 7. Explain to the class that there weren t actually two sticks but that Eratosthenes simply noticed that no structures in Cyene such as pillars cast a shadow at solar noon (which is when the sun is at its highest point in the sky for a particular day) on the Summer Solstice (on which day the noontime sun is higher in the sky than any other day of the year). He carefully measured the length of a shadow in Alexandria for an object of known height at noon on the same day. 8. Hand out take home worksheet. Discussion: To aide you in your discussion you may want to use the overhead (Figure 3) of the map, which will help illustrate the error in pacing out the distance between Cyene and Alexandria and that Alexandria and Cyene are not on the same meridian. Today we know the circumference of the Earth is 40,008 km, where does the error come from? Engage in class discussion about the errors that may occur in Eratosthenes method. Discuss ways we could find the circumference in this day and age. What was the hardest step in finding using this information to find the circumference of the Earth? Ancient Measurements You may notice that the measurement of the stick and the shadow are not in centimetres but in cubits. This was the measurement used in Eratosthenes. Although, it is not necessary to convert from cubits to centimetres to solve the problem, students may want to know. A cubit is centimetres. Alternate Angles are Equal You may want to remind the class that Eratosthenes knew that given two parallel lines, the alternate interior angles are equal. If the class is having trouble, ask them where there are two parallel lines in their diagrams? Possible sources of error Include: The measurement of the distance between Cyene and Alexandria was inaccurate - someone paced the distance and there was no straight path Cyene and Alexandria are not on the same meridian - the geometry of Figure 2 assumes the cities are north/south of each other. If the cities are on different longitudes, the distance measurement between overesti mates, giving a larger estimate of the circumference. Imagine, for example, that instead of Alexandria, Eratosthenes had used another city far away in West Africa, but on the same latitude as Alexandria. At noon, the shadow s angle would still be 7 degrees, but the distance between the cities would be several thousand kilometres, providing an enormous estimate of the circumference. A good way to illustrate this to the class would be to use a globe and lamp. Measuring the length of a shadow precisely can be difficult - hard to see a distinct edge. The sun s rays are not perfectly parallel - the sun is a fixed point very far away but not infinitely far away. Measuring Shadows with Your Class There are online projects where you repeat Eratosthenes experiment with your class by measuring shadows and partnering with another school in a different part of the world. The process is a bit more involved and will take a couple class periods. To find out more visit: curriculum/noonday/index.html

6 Figure 1 - Information for Finding Circumference of the Earth Eratosthenes assumed : The earth is a sphere. Therefore a stick planted vertically in the ground will be perpendicular to the earth. The sun is very far away. When its rays hit the Earth they are parallel. Alexandria is directly north of Cyene they lie on the same meridian. Eratosthenes knew: At noon in Cyene, the sun is directly overhead and casts no shadow. The distance from Cyene to Alexandria is approximately 800km. The stick used in Alexandria measures 4 cubits long and its shadow at noon is cubits.

7 Figure 2 - Eratosthenes Diagram

8 Figure 3 - The Distance Between Cyene and Alexandria