PROLOGUE
The Summer Solstice Experiment
Eratosthenes was the chief librarian of the Great Library at Alexandria. One morning, while he was among the Astronomy scrolls, he overheard a traveler's tale that caught his attention. The traveler had just returned from Syene, far to the south, and spoke of a curious well there.
"At noon on the longest day," the traveler said, "the Sun shines straight down to the water. No shadow at all. You could see your face reflected at the very bottom."
Eratosthenes knew that here in Alexandria, there were always shadows at noon. Even on the summer solstice, the Sun never climbed to the exact zenith. Every column, every obelisk cast its short midday shadow northward.
‘Hold on a moment’, Eratosthenes thought to himself.
‘Some people still think the Earth is flat, but if it were, the Sun should strike both cities at the same angle. But if the Earth were curved—a sphere, Pythagoreans had long claimed—then the Sun's rays, coming in parallel from such a vast distance, would strike at different angles!’
When the summer solstice came, Eratosthenes had prepared. He had placed a vertical post—a gnomon—in the courtyard. As the Sun reached its highest point, he carefully measured the shadow it cast. Using the gnomon's height and the shadow's length, he calculated the angle: about one-fiftieth of a full circle.
One-fiftieth of the curve between Alexandria and Syene.
Now he needed the distance. The royal surveyors kept records—caravans took about 50 days to walk from Alexandria to Syene, covering roughly 100 stadia per day. Call it 5,000 stadia.
If 5,000 stadia represented one-fiftieth of Earth's circumference, then the whole Earth must measure 250,000 stadia around.
Eratosthenes sat back stylus in hand, and contemplated what he had done. With nothing but a stick, a shadow, and geometric reasoning, he had measured the circumference of the Earth.
The Margin of Error
Astronomers today know the circumference of Earth to be 40,075 kilometers.
In Alexandria, the stadion measured around 157.5 meters.
250,000 stadia converted to kilometers then, is 39,375 kilometers.
In 240 BCE, Eratosthenes was within 2% of the value modern astronomers use for the circumference of the Earth using a stick, a shadow, and math.