Mathematics to the Moon: Eratosthenes and Lunar Distance

Introduction: A Promise to Reach the Moon

Once upon a time, Alexander, a young man from Athens, fell in love with a local girl named Adrianna, whose beauty surpassed 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 he could deliver the moon, but he could begin to convince her of his intelligence and cleverness by measuring the distance from the Earth to the Moon. Alexander had long heard stories about his Greek ancestors, who were experts in mathematics and astronomy, so he sought out some wise elders to learn more.

Alexander spent time with two elders: one who 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 and the Circumference of the Earth

Eratosthenes had lived for years in Alexandria, Egypt, and was fond of 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 sunlight was reflected from the water at the bottom of the well. He had never seen this before, and after giving it some thought, realized that on that particular day of the year, the sun was positioned exactly vertically overhead. Interestingly, because Eratosthenes was an observant fellow, he also knew that the sun never achieved a perfectly vertical position over Alexandria. The following year — precisely on June 21 — Eratosthenes measured the sun's reflection and found that it angled off by roughly 7.2 degrees. He made these measurements cleverly by observing the shadow of a stick. Alexander could tell that Eratosthenes was exceptionally proud of this multi-year accomplishment, and he expressed his admiration for the elder's work accordingly.

Eratosthenes continued his explanation. 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 midday on June 21 between the two locations would provide the numbers needed to calculate the circumference of the Earth. The key to this calculation, the elder explained, was recognizing that the vertical sticks used to gauge the angle of sunlight could not be truly parallel, because the Earth is spherical. This means that the two sticks — one in Alexandria and one in Syene — were not parallel to each other. However, the rays of sunlight reaching both locations were parallel to one another. Furthermore, the sun's rays falling on Syene were parallel to the stick there — as Eratosthenes reminded Alexander, the sun cast no shadow on June 21 in Syene.

According to his measurements, Eratosthenes determined 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. He pointed out to Alexander that this angle was maintained from the center of the Earth to its surface, so the distance around the Earth could be derived as 250,000 stades (approximately 23,300 miles). Note that contemporary estimates place Earth's circumference at about 25,000 miles, but varying interpretations of the length of a stade affect how closely Eratosthenes' result aligns with modern figures — in any case, his method was geometrically sound and his result impressively accurate for the era.

Conclusion: Mathematics Across the Ages

Alexander was duly impressed by what he had learned from the two elders, and was confident that Adrianna would be as well. The ancient Greeks had demonstrated that careful observation, proportional reasoning, and geometric thinking could unlock some of the universe's most fundamental measurements — without telescopes, satellites, or computers. The methods of Eratosthenes and Aristarchus stand as enduring testaments to the power of mathematics, and their results remain astonishingly close to those obtained by modern science. For Alexander — and for anyone who has ever looked up at the Moon and wondered — the answer was always there, waiting to be reasoned out.