The determination of the Earth’s true size presented a challenge to ancient mathematicians and astronomers. Philosophers had established the planet’s spherical nature centuries earlier, but translating that understanding into a precise, measurable figure remained elusive. This undertaking required astronomical observation, geometric understanding, and accurate linear measurement over vast distances. The solution demanded a novel approach using simple tools to capture a fraction of the planet’s curvature, allowing the full circumference to be calculated.
Identifying the Ancient Geographer
The person credited with first successfully measuring the Earth’s size using sun angles was Eratosthenes of Cyrene (c. 276 to 195 BCE). Eratosthenes was a renowned polymath, distinguishing himself as a geographer, mathematician, and astronomer. He served as the chief librarian at the Library of Alexandria in Egypt for over four decades. This scholarly environment provided him the resources to devise his elegant experiment, utilizing fundamental geometric principles.
The Essential Setup: Syene and Alexandria
Eratosthenes’ method relied on observations made in two Egyptian cities: Syene (modern Aswan) and Alexandria. In Syene, at noon on the summer solstice, sunlight shone directly to the bottom of a deep well, indicating the sun was precisely at the zenith, or directly overhead. This meant a vertical object in Syene cast no shadow at that exact moment.
Alexandria was situated significantly north of Syene. At the same time on the solstice, a vertical measuring rod, known as a gnomon, was used to observe the sun’s angle. In Alexandria, the gnomon cast a noticeable shadow, proving the sun was not directly overhead. Eratosthenes assumed the sun was so distant that its incoming rays could be treated as parallel when they reached Earth.
Translating Angles to Distance
The shadow in Alexandria allowed Eratosthenes to measure the angular difference between the two locations. This angle was calculated to be about 7.2 degrees, which is exactly one-fiftieth of a full 360-degree circle. Using the geometric principle of alternate interior angles, Eratosthenes realized the shadow angle was equal to the central angle subtended by the arc connecting Syene and Alexandria at the Earth’s center. This meant the physical distance between the two cities represented one-fiftieth of the Earth’s total circumference.
The linear distance between Syene and Alexandria was measured by specialized surveyors called bematists (professional paces-counters). They estimated the distance to be 5,000 stadia, the common unit of length used then. Eratosthenes completed his calculation by multiplying the measured distance by 50. Multiplying 5,000 stadia by 50 yielded a result of 250,000 stadia for the Earth’s circumference.
Evaluating the Historical Accuracy
Eratosthenes’ final measurement demonstrated remarkable accuracy, given the limitations of his ancient tools and assumptions. The accuracy depends on the length assumed for the stade, a unit that varied across the Greek world. If the Egyptian stade (about 157.5 meters) is used, his result converts to roughly 39,375 kilometers, an error of less than two percent compared to the modern polar circumference of 40,008 kilometers.
If a longer Greek or Olympic stade (approximately 185 meters) is used, the calculation yields a larger error. Minor inaccuracies also existed, such as Syene not being precisely on the Tropic of Cancer, and the cities not lying exactly on the same north-south meridian line. Despite these factors, the method was sound, providing the first reasonably accurate size of our planet.