Size of the Earth

You can measure the size of the earth with two shadow sticks. Eratosthenes of Cyrene had already considered and realised this in 220 BC. He was followed by Poseidonius (around 100 BC) and others.

We simply measure the distance b of the two places on earth A and B in a unit of length (in ancient times this would be stadia, in modern times perhaps kilometres). Then we can use the rule of three to determine the circumference of the earth from the angular difference between the locations:

The arc segment b is the absolute distance between locations A and B, which should be on the same longitude. For long distances, it is determined from the travelling speed v with b = v * t, where t is the duration of the journey.

The only question is, how do you determine the angular difference |α – β| between the two locations?

Eratosthenes’ approach: Measure the noon altitude of the sun on the same day. The difference in the position of the sun corresponds to the angular difference between the locations.

Observation

Two observers at different latitudes, e.g. Athens and Berlin, measure the midday height of the sun on the same day using a shadow stick.

A glass bottle or an acacia spike can be used as a shadow stick – depending on what is available at the time.

The spirit level checks whether the ground is really level.
It is best to mark the end of the shade with the pencil and measure the length later at your leisure.

If you want to do this as accurately as possible, you should measure the length of the shadow for a whole day from sunrise to sunset. As the change in the height of the sun varies very little around true noon (the derivative of the sine is the cosine and this is zero at the maximum of the sine), true noon can be easily determined by taking the average of many values in the morning and afternoon.

You have measured the height h of the shadow stick and the length l of the shadow.

Example Results (1997)

In 1997, I determined the circumference of the earth using this method to be 39,300 km, which deviates from the modern table value by only -1.75 %. It deviates by +6.5 % from the historical value that Eratosthenes determined for the earth’s circumference.

Antonina Böhme took the measurement in Berlin while I was on a school trip to Greece. She measured a midday altitude of the sun of approx. 38°, while I measured approx. 50° in Greece.

Over the course of a year, the midday height of the sun fluctuates: see results above.