## Eratosthenes Measures the Earth

When did humans first learn the size of the Earth? Or, for that matter, when did it become clear that the Earth was a sphere?

To those not attuned to science history, it may come as a surprise to learn that the Earth’s size and shape were settled 2,200 years ago, and by the same person. In what was one of the great intellectual feats of antiquity, the Greek Eratosthenes  took the measure of the Earth using only simple tools and mathematics, together with his creative mind. This post recounts how Eratosthenes did it.

The Great Calculation

The story begins in 230 BC, at the Library of Alexandria, where Eratosthenes was the librarian. In a papyrus book, he read something that seemed quite odd. The book said that at a place called Aswan, far to the south of Alexandria, the Sun cast no shadow at noon on the longest day of the year (i.e., at the Summer Solstice). The Sun was so precisely vertical that its reflection could be seen in the water at the bottom of a deep well.

Eratosthenes knew that the same phenomena did not occur at Alexandria.  On the next solstice, he verified this by measuring the Sun’s angle at noon, and he found that the Sun was slightly more than 7 degrees from vertical.  Pondering this, Eratosthenes saw that the difference could be explained if the Earth was curved instead of flat (see the diagram).  Unlike at Aswan, a vertical stick at Alexandria was not pointed directly at the Sun, and this caused it to have a shadow.

This was good evidence that the Earth was round, but Eratosthenes saw that he could also compute the Earth’s size. In the drawing, note that 7 degrees is also the angle formed by the two Earth radii. This follows from simple geometry: the two 7 degree angles are what we call “alternate interior angles” of the two parallel lines formed by the Sun’s rays.

With a circle being 360 degrees, this means that the distance from Alexandria to Aswan is 7 /360 of the Earth’s circumference. More to the point:

Circumference = 360/7 x (Alexandria to Aswan distance).

No one knew the distance to Aswan with any precision, so Eratosthenes set about finding the distance. He hired a man to walk the full distance, counting the number of steps. This was no small thing; the distance turns out to be 800 km!.

The calculated circumference was then 39,690 km. This is within 2% of the correct value, and it was 1,800 years before any one made a more accurate measurement. An impressive result indeed. And, as Carl Sagan put it, all Eratosthenes needed was sticks, feet, and brains.