Saturday, February 21, 2015

Earth Size

In 240 BC Eratosthenes estimated the circumference of the Earth to within 100 miles.
[Eratosthenes] ... is a mathematician among geographers, and yet a geographer among mathematicians; and consequently on both sides he offers his opponents occasions for contradiction.
Eratosthenes (276–194 BC)
Ancient Greeks knew Earth was spherical. How did Eratosthenes accurately estimate its circumference?

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°.

Two Cities

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.

stade is an ancient Greek unit of measurement equivalent to the distance of footraces in the ancient Olympic Games in Olympia.
1 stade ~ 600 feet
Using 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:
7.2°=4,400 stades
Solving for circumferential distance C:
C=360°   x 4,400 stads  = 220,000 stads or 25,000 miles
Remarkably Eratosthene's estimate overshot the circumference of the Earth by a mere 99 miles.
25,000 miles24,901 miles