Earth, planets and orbits
c. 240 BC
Measuring the Earth with a shadow
Eratosthenes of Cyrene
Can you measure the whole planet without leaving town?
At noon on midsummer’s day, the Sun shone straight down a well in Syene, in southern Egypt. In Alexandria, further north, a pole still cast a shadow, tilted by about 7.2 degrees: one-fiftieth of a full circle. So the Earth must be about fifty times the distance between the two cities around.
First, make a guess
If the Earth were flat and the Sun very far away, how would the two sticks’ shadows compare at noon?
- Shadow angle at Alexandria
- 7.20°
- Eratosthenes’ reasoning gives
- 40,000 km around
- Calculation
- 50.0 × 800 km
Eratosthenes assumed the Sun is so far away that its rays arrive parallel. Then the only way two sticks can cast different shadows is if the ground between them curves. A flat Earth with a nearby Sun could fool you, which is why the assumption matters. The angle on the globe is drawn larger than life.
How they figured it out
Eratosthenes ran the great Library of Alexandria. He heard that in Syene, far to the south, the noon Sun on midsummer’s day shone straight down a deep well, with no shadow at all. In Alexandria, on the same day, an upright pole still cast a short shadow.
He made two bold assumptions: the Sun is so far away that its rays arrive parallel, and Syene lies roughly due south. Then the shadow’s angle, about a fiftieth of a circle, must equal the angle between the two cities at the centre of the Earth. So the Earth’s circumference is fifty times the distance between them. Nobody knows exactly how long his “stadion” was, so we can’t say precisely how close he got, but it was remarkably close.
The equation
If walking a distance d turns the Sun’s angle by 7.2°, then a full 360° circle around the Earth is fifty times d.
See it in real life
Watch the Pole Star climb
Drive about 111 km due north and the Pole Star sits one degree higher in the sky. 360 of those steps take you all the way around: about 40,000 km. Schools in different cities still repeat Eratosthenes’ experiment together on the equinox, comparing stick shadows over the internet.
Where engineers use it
Surveying and GPS
Measuring angles between known points to find distances (triangulation) mapped whole countries and still checks GPS.
Arc length
Arc is how engineers size belts, gears and curved roads.
Worked example: His numbers, in kilometres
- Angle: of a circle.
- Distance between cities: about 800 km.
- Circumference km. Today’s value: 40,075 km at the equator.
Where it came from
- Euclid’s Elements c. 300 BC
Built on
Eratosthenes used Euclid’s rule for angles across parallel lines to turn a shadow into the size of the Earth.