Theorem: Parallel lines cut transversals into proportional segments. If parallel lines are intersected by two transversals, then the ratios of the segments determined along each transversal between the parallel lines are equal.
- The word geometry is Greek for geos (earth) and metron (measure). A Greek named Eratosthenes, who lived over years ago, used geometry to measure the circumference of the earth. The Greeks knew the earth was a sphere. They also knew the sun was so far away that rays from the sun (as they met the earth) were, for all practical purposes, parallel to each other. Using these two facts, here is how Eratosthenes calculated the circumference of the earth.
- Eratosthenes lived in Egypt. He calculated the circumference of the earth without ever leaving Egypt. Every summer solstice in the city of Syene, the sun was directly overhead at noon. Eratosthenes knew this because at noon on that day alone, the sun would reflect directly (perpendicularly) off the bottom of a deep well.
- On the same day at noon in Alexandria, a city north of Syene, the angle between the perpendicular to the ground and the rays of the sun was about 7.2 degrees. We do not have a record of how Eratosthenes found this measurement, but here is one possible explanation:
- Imagine a pole perpendicular to the ground (in Alexandria), as well as its shadow. With a tool such as a protractor, the shadow can be used to determine the measurement of the angle between the ray and the pole, regardless of the height of the pole.
If the height of the pole is 10 meters, what would the length of the shadow be?
- You might argue that we cannot theorize such a method because we do not know the height of the pole. However, because of the angle the rays make with the ground and the 90 degree angle of the pole with the ground, the triangle formed by the ray, the pole, and the shadow are all similar triangles, regardless of the height of the pole. Why must the triangles all be similar?
- This measurement of the angle between the sun’s rays and the pole was instrumental to the calculation of the circumference. Eratosthenes used it to calculate the angle between the two cities from the center of the earth.
- You might think it necessary to go to the center of the earth to determine this measurement, but it is not. Eratosthenes extrapolated both the sun’s rays and the ray perpendicular to the ground in Alexandria. Notice that the perpendicular at Alexandria acts as a transversal to the parallel rays of the sun, namely, the sun ray that passes through Syene and the center of the earth, and the sun ray that forms the triangle with the top of the pole and the shadow of the pole. Using the alternate interior angles determined by the transversal (the extrapolated pole) that intersects the parallel lines (extrapolated sun rays), Eratosthenes found the angle between the two cities to be 7.2 degrees.
- How can this measurement be critical to finding the entire circumference?
- Eratosthenes divided 360 by 7.2, which yielded 50. So, the distance from Syene to Alexandria is 1/50 of the circumference of the earth. The only thing that is missing is that distance between Syene and Alexandria, which was known to be about stades; the stade was a Greek unit of measurement, and 1 stade is about 600 feet.
- So, Eratosthenes’ estimate was about 50 x 5000 x 600 feet, or about 28,400 miles. A modern-day estimate for the circumference of the earth at the equator is about 24,900 miles.
- It is remarkable that around 240 B.C.E., basic geometry helped determine a very close approximation of the circumference of the earth.
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