The Solar Geometry

Measuring the Earth with Shadows and Strides

Among the hundreds of thousands of papyrus rolls cataloged in the Great Library of Alexandria around 240 BCE was a routine administrative report from the southern frontier garrison of Syene, modern Aswan, near the First Cataract of the Nile. The scroll noted a local phenomenon observed every year on the twenty-first of June, the summer solstice. At exact local solar noon on that single day, vertical temple columns cast no shadow at all, and the sun shone straight down into the deepest stone water wells, illuminating the water at the bottom without touching the masonry walls.

For most scholars in Hellenistic Egypt, the report was merely a traveler's curiosity. While educated Greeks since Aristotle and Pythagoras agreed that the Earth was a sphere—pointing to the circular shadow cast by the Earth across the Moon during lunar eclipses and the way ships disappeared hull-first below the horizon—the physical size of the globe remained a matter of guesswork. Philosophers offered rough assertions ranging from 300,000 to 400,000 stadia without providing a testable geometric method to verify the figure. Many contemporaries regarded the planetary circumference as a dimension beyond human measurement, accessible only by sailing around the oceans or walking from pole to pole.

Eratosthenes of Cyrene, a Greek polymath whom King Ptolemy III Euergetes had summoned from Athens to serve as third chief librarian of Alexandria, read the Syene record differently. Nicknamed Beta by rival scholars because he worked across geography, mathematics, chronology, and philology rather than claiming first rank in a single specialty, Eratosthenes realized that a shadowless well in Syene provided a fixed radial line drawn straight from the center of the Earth to the sun.

If the sun was sufficiently distant from the Earth, its incoming rays arrived at every point on the planet along parallel tracks. On the next summer solstice at solar noon, Eratosthenes stood in a courtyard in Alexandria and erected a calibrated vertical bronze rod—a gnomon—inside a hemispherical sundial bowl known as a skaphe. At the exact moment when the sun stood directly overhead in Syene and cast zero shadow, the bronze gnomon in Alexandria cast a wedge of shade along the graduated inner scale of the bowl. Eratosthenes measured the angle between the vertical rod and the tip of the shadow. It measured one-fiftieth of a complete circle, or 7.2 degrees:

Using the theorem of alternate interior angles established fifty years earlier by Euclid in Alexandria, Eratosthenes drew two parallel solar rays striking the curved surface of the Earth—one entering the well at Syene and proceeding straight to the Earth's center, the other grazing the tip of the gnomon in Alexandria. Because the gnomon itself pointed straight down toward the Earth's center, the 7.2-degree shadow angle measured in Alexandria was geometrically identical to the central angle subtended by the arc of land between Syene and Alexandria at the core of the planet.

If the overland arc between the two cities represented one-fiftieth of the Earth's full 360-degree cross-section, then the total circumference $C$ of the Earth was fifty times the ground distance $D$ between Alexandria and Syene:

To determine $D$ without relying on irregular caravan timings, Eratosthenes turned to the cadastral records of the Ptolemaic royal bematists—professional surveyor-striders trained to walk the flat floodplains along the Nile with mechanically uniform, calibrated paces to measure distances for taxation after the annual inundation. Their ledgers recorded the meridian distance from Syene north to Alexandria as 5,000 stadia.

Multiplying 5,000 stadia by 50, Eratosthenes calculated the circumference of the Earth at 250,000 stadia, later refining the figure to 252,000 stadia so that dividing the globe into sixty intervals yielded a clean 4,200 stadia per sector. Under the Egyptian itinerant stadion of 157.5 meters, 252,000 stadia equals 39,690 kilometers—within one percent of the Earth's true polar circumference of 40,008 kilometers; under the Attic stadion of 185 meters, it equals 46,620 kilometers, within sixteen percent. A century and a half later, the Stoic philosopher Posidonius recalculated the circumference using the star Canopus and arrived at a smaller figure of 180,000 stadia. When Claudius Ptolemy enshrined that smaller number in his second-century Geography, it shrank the scholarly map of the world by nearly a third—the exact underestimate that persuaded Christopher Columbus in 1492 that Asia lay just three thousand nautical miles west of the Canary Islands.