[Eratosthenes] ... is a mathematician among geographers, and yet a geographer among mathematicians; and consequently on both sides he offers his opponents occasions for contradiction.Ancient Greeks knew Earth was spherical. How did Eratosthenes accurately estimate its circumference?
— Eratosthenes (276–194 BC)
The answer lies in a tale of two cities: Syene and Alexandria. Syene was the ancient name of the city known today as Aswan.
Sun Straight up Syene
Eratosthenes heard that the sun could be observed straight overhead from the bottom of a well in Syene at noon during the summer solstice. No shadows would have been cast on the walls of the well. No shadows would have been cast by the style of an above-ground sundial.
The Alexandria Angle
At noon in Alexandria during a summer solstice, Eratosthenes observed a shadow cast by a column.
Using the height of the column (h) and the distance (d) of the edge of shadow from the column, Eratosthenes used the trigonometric functions tangent and arctangent to determine the shadow angle θ.
Since tan(θ) = d / h, Eratosthenes used the arc-tangent to find the angle.
|The History of Geodesy, NOAA|
θ = arctan(d / h)Eratosthenes found the angle θ of the shadow cast by the column to be 7.2°.
Syene to Alexandria was an established trade route of camel-driven caravans. Eratosthenes knew the distance between Syene and Alexandria was about 4,400 stades.
A stade is an ancient Greek unit of measurement equivalent to the distance of footraces in the ancient Olympic Games in Olympia.
1 stade ~ 600 feetUsing the 7.2° angle θ of the shadow and the distance between Alexandria and Syene, Eratosthenes used an arithmetic ratio. Expressed in words, he recognized
The shadow angle 7.2° is to 360° of the Earth as the 4,400 stades is to the circumferential distance C of the Earth.Expressed as a ratio:
Solving for circumferential distance C:
7.2° = 4,400 stades 360° C
Remarkably Eratosthene's estimate overshot the circumference of the Earth by a mere 99 miles.
C = 360° x 4,400 stads = 220,000 stads or 25,000 miles 7.2°
25,000 miles 24,901 miles
- Eratosthenes, Wikipedia.
- Eratosthenes is Known as the Father of Geography, Matt Rosenberg, About Education.
- June 19, 240 B.C.: The Earth Is Round, and It's This Big, Randy Alfred, Wired Archive, 19 June 2008.
- The History of Geodesy, NOAA.