1 00:00:00,000 --> 00:00:07,000 Did you know that over 2,000 years ago a Greek librarian used geometry to determine the circumference of the Earth? 2 00:00:12,000 --> 00:00:15,000 I did aerostomes and used to find the circumference of the Earth. 3 00:00:15,000 --> 00:00:20,000 What are the angle relationships between parallel lines and a transversal? 4 00:00:21,000 --> 00:00:26,000 The concept of the Earth being a large sphere was not unknown to the ancient Greeks. 5 00:00:26,000 --> 00:00:30,000 An everyday observation, such as the disappearance of ships below the horizon, 6 00:00:30,000 --> 00:00:33,000 indicated that the Earth might be spherical or round. 7 00:00:33,000 --> 00:00:35,000 But how large was it? 8 00:00:35,000 --> 00:00:39,000 The person who figured it out was a librarian named Aristophanes, 9 00:00:39,000 --> 00:00:43,000 who lived in Alexandria, Egypt, about 300 BC. 10 00:00:43,000 --> 00:00:48,000 While looking through a scroll one day, he read that at noon on the longest day of the year, 11 00:00:48,000 --> 00:00:53,000 a vertical column cast no shadow in Cyene, a city south of Alexandria. 12 00:00:53,000 --> 00:00:56,000 Aristophanes knew that this did not happen in Alexandria. 13 00:00:56,000 --> 00:00:58,000 He thought to himself, 14 00:00:58,000 --> 00:01:03,000 how was it possible to have shadows in Alexandria and not in Cyene at the same time of day? 15 00:01:03,000 --> 00:01:10,000 Aristophanes figured out that the sun must be directly overhead in Cyene, but not in Alexandria. 16 00:01:10,000 --> 00:01:15,000 Aha! Here was proof that the Earth's surface is curved. 17 00:01:15,000 --> 00:01:20,000 Using a little geometry, Aristophanes set out to determine the circumference of the Earth 18 00:01:20,000 --> 00:01:22,000 and find out just how big it is. 19 00:01:22,000 --> 00:01:24,000 Just like our pizza example, 20 00:01:24,000 --> 00:01:30,000 if our friend Aristophanes could determine the central angle at the center of the Earth 21 00:01:30,000 --> 00:01:33,000 and the length of the edge or arc, 22 00:01:33,000 --> 00:01:36,000 then he could figure out the circumference of the Earth. 23 00:01:36,000 --> 00:01:41,000 Now, finding the length of the edge or arc was fairly simple math. 24 00:01:41,000 --> 00:01:46,000 Aristophanes asked a friend to walk from Alexandria to Cyene 25 00:01:46,000 --> 00:01:49,000 to measure the distance between the two cities. 26 00:01:49,000 --> 00:01:56,000 His friend estimated the distance to be around 800 kilometers or about 500 miles. 27 00:01:56,000 --> 00:02:00,000 Finding the central angle, however, would take some geometry. 28 00:02:00,000 --> 00:02:04,000 First, Aristophanes assumed correctly, I might add, 29 00:02:04,000 --> 00:02:08,000 that the sun's rays are parallel since the sun is so far away. 30 00:02:08,000 --> 00:02:09,000 Check this out. 31 00:02:09,000 --> 00:02:12,000 In this diagram, we can see that there is no shadow at Cyene, 32 00:02:12,000 --> 00:02:15,000 while there is a shadow in Alexandria. 33 00:02:15,000 --> 00:02:21,000 The line that is formed by the gnomon, or vertical column at Alexandria and the center of the Earth, 34 00:02:21,000 --> 00:02:27,000 cuts or intersects the two parallel lines formed from the sun's rays. 35 00:02:27,000 --> 00:02:31,000 A line that intersects two parallel lines is called a transversal. 36 00:02:31,000 --> 00:02:35,000 The two angles formed from the transversal line and the parallel lines 37 00:02:35,000 --> 00:02:38,000 are called alternate interior angles. 38 00:02:38,000 --> 00:02:42,000 And, according to geometric rule, they are equal. 39 00:02:42,000 --> 00:02:43,000 Let's prove it. 40 00:02:43,000 --> 00:02:48,000 Take a piece of paper of any width and draw a diagonal line on it. 41 00:02:48,000 --> 00:02:51,000 Label the angles A and B just like this. 42 00:02:51,000 --> 00:02:56,000 Now, cut the paper along the diagonal so you have two triangles. 43 00:02:56,000 --> 00:03:00,000 Compare angles A and B by placing one angle on top of the other. 44 00:03:00,000 --> 00:03:02,000 Hey, what do you notice? 45 00:03:02,000 --> 00:03:05,000 The angles are equal no matter what size paper you started with. 46 00:03:05,000 --> 00:03:06,000 Right. 47 00:03:06,000 --> 00:03:10,000 When two parallel lines are intersected by a transversal, 48 00:03:10,000 --> 00:03:14,000 the alternate interior angles are equal. 49 00:03:14,000 --> 00:03:17,000 Huh, Aristophanes was quite a geometer. 50 00:03:17,000 --> 00:03:24,000 From his measurements, Aristophanes calculated the sun's rays made an angle of 7.5 degrees at Alexandria. 51 00:03:24,000 --> 00:03:29,000 Now, since this angle was formed by two parallel lines and a transversal, 52 00:03:29,000 --> 00:03:33,000 the central angle of the Earth must also be 7.5 degrees. 53 00:03:33,000 --> 00:03:39,000 By knowing these two things, the central angle and the distance from Alexandria to Syene, 54 00:03:39,000 --> 00:03:42,000 Aristophanes calculated the circumference of the Earth. 55 00:03:42,000 --> 00:03:51,000 360 degrees divided by 7.5 degrees equals 48 slices of the Earth. 56 00:03:51,000 --> 00:03:53,000 Are you still with me? 57 00:03:53,000 --> 00:03:54,000 Okay, hang tight. We're almost there. 58 00:03:54,000 --> 00:04:00,000 Now, if you remember that the estimated distance between Alexandria and Syene is 800 kilometers, 59 00:04:00,000 --> 00:04:06,000 and you multiply that distance by the number of slices in the Earth, 48, 60 00:04:06,000 --> 00:04:10,000 what is the circumference of the Earth? 61 00:04:10,000 --> 00:04:17,000 Well, if you estimated that distance to be 38,000 kilometers, you're absolutely right. 62 00:04:17,000 --> 00:04:25,000 Aristophanes' estimate was really close to the Earth's circumference, which is 40,074 kilometers. 63 00:04:25,000 --> 00:04:32,000 His percentage error was about 5%, and was probably due to an error in the distance between the two cities. 64 00:04:32,000 --> 00:04:35,000 5%? Huh, that's pretty good, 65 00:04:35,000 --> 00:04:42,000 considering Aristophanes used only his feet, his eyes, his imagination, and, of course, his knowledge of geometry.