Eratosthenes and the Circumference of the Earth
Date: 10/7/95 at 23:8:58 From: Tucker - Joanne Subject: Ask Dr. Math Dear Dr. Math, How did Eratosthenes measure the circumference of the earth? Thanks, Our Third Grade Class
Date: 10/10/95 at 16:53:35 From: Doctor Andrew Subject: Re: Ask Dr. Math Well, according to the Encyclopedia Americana and the Encyclopedia Britannica, Eratosthenes observed that at noon on the summer solstice (the longest day of the year) the sun was directly overhead in the city of Syene in Egypt (it is called Aswan now). I've heard elsewhere that he knew this because at that time, no shadow was cast in a well. A well isn't necessary to observe this though, since any container with parallel walls such as a box or a tube will not have a shadow when light comes from directly above it. Try it out yourself. He then assumed that the sun was so far off that its rays hit the earth in parallel. If you imagine all the lines from the surface of one ball to another, you can see that as the balls get further apart, all the lines become nearly parallel. Parallel means that the lines all go in exactly the same direction. He also assumed that the earth was shaped like a ball. He also knew that Syene was on the same meridian as the city of Alexandria. The earth is a ball that spins around a line called its axis. A meridian, (also called a line of longitude) is a line on the surface of the earth from one end of this axis to the other. I'm not sure how he knew these two cities were on the same meridian; maybe he knew that the sun set at the same time when it was directly between the two cities. If you follow a meridian all the way around the earth you get a circle, like the equator. Finally, Eratosthenes knew that the distance between Alexandria and Syene was 5000 stadia, a Greek unit for measuring length. This was about 500 miles. So, on the summer solstice, at noon, in Alexandria, Eratosthenes measured the angle of the sun's rays. You could do this by finding the angle at which a shape casts the least shadow. Suppose you had a globe that had a metal band around it that could rotate around the globe but could also be completely horizontal (globes usually have bands around them that are vertical). You may have one like this in your classroom. If you take the globe out and then rotate this band until its shadow is only a line, it will be parallel to (in the same direction as) the rays of the sun. Think about what fraction of a whole circle you had to rotate the band. Well, Eratosthenes probably had a device similar to this which he had to rotate 1/50 of a whole circle to get it to line up with the sun's rays. Using a little geometry (that is a little tough for 3rd grade) he then knew that 5000 stadia was 1/50 of the circumference of the earth. This means that he needed to use 50 of these lengths to surround the earth. So he multiplied 5000 by 50 to get 250,000 stadia. Then he added 2000 more to make up for what he thought were bad measurements. So he calculated the circumference of the earth to be 252,000 stadia. We know that the distance between Alexandria and Syene is about 500 miles, so using his fraction 1/50, we can get the circumference of the earth to be about 500 * 50 = 25000 miles, which is about right. Since historians aren't sure how long one stadia is, we haven't been able to figure out how close Eratosthenes was to the correct answer, but we do know that the way he tried to solve them problem was correct. In those days they couldn't easily make very accurate measurements of the distance between two places, so this could cause a lot of error. I hope this is all clear, but there are probably some messy points. If you have any questions about this, please send them to us. If you want to know how Eratosthenes used Geometry to show that the 1/50 of a circle on the angle measuring device means that the distance between the two cities was 1/50 of the circumference of the earth, I'd be glad to try to explain it. -Doctor Andrew, The Geometry Forum
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