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Culture and history · The sky-measurers

Eratosthenes: measuring the Earth with a shadow

In the 3rd century BC, a scholar in Alexandria worked out the Earth's circumference from the difference between the shadows in two cities. Calculate for yourself how parallel rays and alternate interior angles made this possible, and which numbers are documented and which are debated.

  • 18 min
  • 2 simulations
  • Mini quiz

Updated:

Preview

The question

At noon on the summer solstice, the longest day of the year, an upright stick in Syene (modern Aswan), on the banks of the Nile, casts almost no shadow. At the same moment, in Alexandria, about 800 kilometres to the north, a stick of the same height casts a short but clear shadow. Same Sun, same sticks. So where does this small difference in the shadows come from? And how, about 2,250 years ago, did a librarian work out the size of our planet from it?

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No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Eratosthenes

The Earth's circumference in stadia

Move the circumference slider. In the drawing, the angle that the arc between the two cities makes at the centre of the Earth changes too. When this central angle equals the shadow angle measured in Alexandria, press Check.

Prediction

Predict first

If the Earth were bigger than it is, what would happen to the difference in shadow angle between two cities 5,000 stadia apart?

It doesn’t matter whether you’re right; the simulation opens once you choose.

Eratosthenes

Measuring from Sinop and Mersin

The simulation works with the difference between the two measurements and lights Mersin from directly overhead, as if the Sun were there. This drawing doesn't change the result: what matters is the difference in angle between the two cities. Set the circumference slider, then press Check.

Prediction

Predict first

For this method to work, in which city does the Sun have to be directly overhead?

It doesn’t matter whether you’re right; the simulation opens once you choose.

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Frequently asked

5 questions

Did Eratosthenes discover that the Earth is round?

No. Educated Greeks knew this at least a century before him; Aristotle had written that the Earth's shadow on the Moon during lunar eclipses is always round. Eratosthenes assumed the Earth was round and measured its size.

Did people in the Middle Ages think the Earth was flat?

No. Throughout the Middle Ages, educated people in both the Islamic world and Europe knew the Earth was a sphere; their astronomy books were built on a round Earth. The idea that "everyone in the Middle Ages thought the Earth was flat" is largely a myth that spread in the 19th century.

If Eratosthenes' book is lost, how do we know about his measurement?

We learn his method from the textbook of the astronomer Cleomedes, who lived a few centuries after him. Later writers such as Strabo and Pliny report his result as 252,000 stadia. So what we have is a retelling: some numbers are written down, others are round estimates.

Did Eratosthenes find the Earth's circumference to within 1 per cent?

We cannot say for sure, because the length of a stadion is debated (roughly 157 to 185 m). Depending on that, his result comes to about 39,000 to 46,000 km; the true circumference is about 40,008 km. What is certain is that he found the right order of magnitude.

Does the Sun have to be directly overhead in one city for this method to work?

No. If two observers on the same meridian measure the shadow angle at noon on the same day, the difference between the two angles gives the central angle between the cities. For example, at noon on an equinox you measure about 42.0° in Sinop and about 36.8° in Mersin; the difference is 5.2°.