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1.1 Eratosthenes Measures the Earth (Copyright: Bryan Dorner All Rights Reserved)

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1.1 Eratosthenes Measures the Earth (Copyright: Bryan Dorner All Rights Reserved) 1.1 Eratosthenes Measures the Earth (Copyright: Bryan Dorner all rights reserved) How big is the Earth? The mathematician Eratosthenes (276-195 BCE) lived in the city of Alexandria in northern Egypt near the place where the Nile river empties into the Mediterranean. Eratosthenes was chief librarian at the Alexandria museum and one of the foremost scholars of the day - second only to Archimedes (who many consider one of the two or three best mathematicians ever to have lived). The city of Alexandria had been founded about one hundred years earlier by Alexander the Great whose conquests stretched from Egypt, through Syria, Babylonia, and Persia to northern India and central Asia. As the Greeks were also called Hellenes, the resulting empire was known as the Hellenistic empire and the following period in which Greek culture was dominant is called the Hellenistic age. The Hellenistic age saw a considerable exchange of culture between the conquering Greeks and the civilizations of the lands they controlled. It is from about this period that the predominantly geometric mathematics of the Greeks shows a computational aspect borrowed from the Babylonians. Some of the best mathematics we have inherited comes from just such a blend of contributions from diverse cultures. Eratosthenes is known for his simple but accurate measurement of the size of the earth. The imprint of Babylon (modern Iraq) as well as Greece can be seen in his method. He did not divide the arc of a circle into 360 parts as the Babylonians did, but into 60 equal parts. Still, the use of 60 reveals the influence of the Babylonian number system - the sexagesimal system - which was based on the number 60 in the same way ours is based on the number 10. His method also uses simple ideas from Greek geometry. The sexagesimal system became increasingly the standard for astronomical calculations. The complete Babylonian system of degrees, minutes, seconds, etc. was introduced into Greek mathematics by Hipparchus some 75 years after Eratosthenes. We will have more to say about Hipparchus later. Through the Greek influence on Europe, we have adopted their adoption and still commonly measure angles in degrees and minutes some two thousand years after Eratosthenes and Hipparchus - as had been done in Babylon for some two thousand years before their time. The story goes that Eratosthenes knew that on noon of the longest day of the year (the summer solstice) the sun was straight above the town called Syene, for travelers reported the curious fact that a vertical pole cast no shadow at that time in that town. Eratosthenes reportedly checked by traveling to Syene and observing that, indeed, the sun shone straight down to the bottom of a deep well at noon on the summer solstice. (Today the spot is called Aswan and is the location of a huge dam on the Nile river.) However in Alexandria, which lies pretty much due north of Syene, Eratosthenes measured the angle of the sun's rays at noon on the summer solstice to be (in "modern" units) 7o 12'. Now the distance from Alexandria to Syene was known to be about 5000 stades. We can easily complete Eratosthenes' determination of the size of the earth in the following steps: Exercise: 1. First note that the angle at the center of the earth determined by Alexandria and Syrene is equal to the angle a measured by Eratosthenes. Explain why the angles are equal. M P 2. A proportion is an equation of the form: N = Q . Write a proportion that relates the following four quantities: d, the distance bewteen Alexandria and Syene, c, the circumference of the Earth, a, the angle that Eratosthenes measured, and 360o. 3. Use the result of the previous exercise to find the circumference of the earth in stades. Unfortunately, more than one distance called "stade" was in use in the ancient world and there is some controversy about which one was used. Exercise 4. If the Egyptian stade of about 516.7 feet was used, compute the circumference of the earth in miles. Find a reference book (visit the library if necessary - but the dictionary may suffice) that has the modern measurement of the circumference of the earth and compare the two values. The result of your computation should be pretty close to the modern value. It is of interest to note that this measurement was known to the learned folk of Columbus' time and is one of the reasons he found it difficult to get financial backing for his voyage west. Indeed, if there were no American continents and only open ocean between Europe and Asia, Columbus very likely could not have completed the voyage in the vessels of his day. He thought the world was much smaller than it really is. Eratosthenes' Data: Eratosthenes' measurement rests on two pieces of data: the distance from Alexandria to Syene and the angle he measured at Alexandria on the summer solstice. We'll talk about each in turn. Exactly how Eratosthenes determined the distance from Alexandria to Syene is apparently not known. As is almost always the case in investigating ancient events, the story has to be pieced together from accounts in manuscripts often written hundreds of years after the event. Daniel Boorstin, in The Discoverers, claims he was told that a camel caravan would travel 500 stades in a day and the trip to Aswan took 10 days while David Burton, in his History of Mathematics, tells us that the distance had been measured by a surveyor or "bematistes" trained to walk with equal strides and count the paces. Eratosthenes seems to have been aware of the approximate nature of the number 5000 as the number of stades from Alexandria to Syene. For one thing the phenomena of vertical poles casting no shadows at the summer solstice was said to occur within a region of 300 stades around Syene. For another, there is evidence that Eratosthenes modified his computed value of 250,000 stades and to have used instead the value 252,000 stades. Presumably he would only have done so if he thought there was some room for error in his computed value. The change from the "round number", 250,000, to 252,000 may seem strange until we realize that 250,000 is a "round number" in the decimal system, but not in the Babylonian sexagesimal system - in particular 250,000 is not evenly divisible by 60 while 252,000 is. This is further evidence of the influence of Babylonian numerical methods on Eratosthenes' work. Exercises: 5. Assuming that vertical poles cast no shadows on the summer solstice for a distance of 300 stades around Syene, find the largest and smallest possible circumference for the Earth in stades using Eratosthenes' method. Is his assumed value of 252,000 within the likely limits of error? 6. If the value of 252,000 stades were indeed the true value of the circumference of the earth, and Eratosthenes' angle measurement was correct, how many stades are there from Alexandria to Syene? 7. The original definition of a nautical mile was the length of arc of the earth's cicumference subtended by an arc of an angle at the center of the earth of one (ordinary) minute. Find the circumference of the earth in nautical miles. 8. Use a modern value of the circumference of the earth in (ordinary) statute miles to calculate the difference between a nautical mile and a statute mile in feet. 9. Is the number 252,000 really divisible by 60? Now we turn to Eratosthenes' measurement of the angle between the sun's rays and a vertical pole at Alexandria at the summer solstice. One source we have for Eratosthenes' measurement is Cleomedes who probably wrote in the middle of the first century B.C - over 150 years after the measurement. According to this account, Eratosthenes measured the angle using a sundial in the shape of a bowl. If a sundial is shaped like a hemisphere and has a vertical pointer (called a gnomon - pronounced "no–mahn") from the bottom to the center of the sphere, then the inside of the bowl represents an inverted map of the dome of the sky and the position of the tip of the shadow of the gnomon shows the position of the sun in the sky. Eratosthenes could easily have used such a device to measure the angle, a , between the sun and the vertical at Alexandria. Sun a Hemispherical Bowl Arc of angle a 1 = __ of circle 50 A similiar, but very large, hemispherical sundial was constructed as part of the Jantar Mantar, a stone observatory constructed in 1724 by Maharaja Jai Singh at Delhi in India. Today the Jantar Mantar (Time Device) is a city park and tourist attraction in the modern capital, New Delhi. Jai Singh constructed similar Jantar Mantars at other spots in India. (The one in Ujain is in particularly good condition and still "works".) The hemispherical "sundial" is but one of several large structures in the Jantar Mantar built for observing the heavens. It is called the Jayaprakash Yantra. The Jayaprakash Yantra actually consits of two complementary bowls. The two hemispherical bowls of the Jayaprakash Yantra: the foreground shows about one fourth of one bowl; the second bowl can be seen in the background. Each bowl contains part of the hemispherical measuring surface in such a way that where one bowl has part of the measuring surface, the other bowl has a space where an observer can walk and vice versa. In this way all portions of the hemisperical surface are easily accessible to the observer in a way that would not be possible using a single hemispherical bowl.
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