Capturing Astronomical Geometry the Greek Way

Mathematics -- to the Moon & Back

Once upon a time, Alexander, a young man from Athens fell in love with a local girl, Adrianna, whose beauty was far greater than any other young woman he had ever seen. Alexander was so smitten with Adrianna that he promised her the moon. Being an astute girl, Adrianna told Alexander that she wasn't at all sure that he could deliver the moon, but he could begin to convince her that he was intelligent and clever by measuring the distance from the earth to the moon. Alexander had long heard the stories about his Greek ancestors who were experts in mathematics and astronomy, so he sought out some wise elders to learn more.

Alexander spent some time with two elders, one of whom told him he knew how to measure the size of the earth (which, Alexander mused, was bound to impress, Adrianna), and another who professed to know how to measure the distance to the moon. Alexander grew more confident that he could win Adrianna's affections. Eratosthenes had lived for years in Alexandria, Egypt, and was prone to taking long walks -- and sometimes, camel rides. He often stopped by a deep well at midday, in the town of Syene, to refresh himself and his camel. On June 21 (in the third century B.C.), Eratosthenes noticed that the sunlight was reflected from the water at the bottom of the well. He had never seen this before, and giving it some thought, realized that on this day of the year, the sun was exactly vertically overhead. Interestingly, because Eratosthenes was an observant fellow, he knew that the sun never achieved a position of being exactly vertical over Alexandria. The next year -- precisely on June 21 -- Eratosthenes measured the refection of the sun and saw that it angled off by roughly 7.2 degrees. Cleverly, he made these measurements by using the shadow of a stick. Alexander could tell that Eratosthenes was exceptionally proud of this multi-year accomplishment, and he suitably expressed his admiration for the elder's work. Eratosthenes continued. The distance from Alexandria to Syene was 5,000 stades (1 stade = 500 feet) to the south. The difference in the angle of sunlight at the midday point on June 21 would give the numbers needed to figure out the circumference of the earth. The key to this calculation, the elder explained, was recognition that the idea of verticality of the sticks used to gauge the angle of sunlight cannot be perfect since the earth is spherical. This means, Eratosthenes explained, that the two sticks -- one in Alexandria and one in Syene -- were not parallel. However, the rays of the sun that reached the two locations were parallel. And further, the rays of the sun falling in Syene were parallel to the stick -- Eratosthenes reminded Alexander to recall that the sun did not create a shadow on June 21 in Syene. According to his measurements, Eratosthenes asserted that the angle between the sunlight and the stick at midday in midsummer in Alexandria was 7.2 degrees, or 1/50 of a complete circle. Eratosthenes pointed out to Alexander that, from this, it should be obvious that the angle would maintain from the center of the earth to the surface of the earth, so the distance around the earth could be derived to be equal to 250,000 stades (or 23,300 miles). [Note that contemporary estimates are 25,000 miles, but that interpretations of the