How Do You Measure the Earth?

A mathematician, astronomer, geographer, philosopher, and poet, he was born in Cyrene, studied in Athens, and was later invited to Alexandria, where he eventually became head of the famous Library of Alexandria.

It was here that Eratosthenes began to ponder a question that seemed almost impossible:

How could anyone find out how big the Earth is?

He had no satellites. No airplanes, modern maps, or precise surveying instruments. He could not travel all the way around the Earth and measure it.

But he had the Sun, a shadow, and mathematics.

A Curious Report from Syene

Eratosthenes knew of a remarkable phenomenon observed in the city of Syene—modern-day Aswan, far to the south of Alexandria.

Around noon on the day of the summer solstice, the Sun there rose almost directly overhead. Vertical objects cast virtually no shadow, and according to the famous account, sunlight reached the bottom of a deep well.

But in Alexandria, something different happened.

On that same day, when the Sun reached its highest point in the sky, a vertical post cast a shadow.

Why?

One Sun. Two Cities. Two Different Shadows.

Imagine two vertical posts.

One stands in Syene.

The other stands in Alexandria.

The Sun is so far from the Earth that the rays reaching it can be treated as parallel.

In Syene, the rays fall almost straight down—so there is almost no shadow.

In Alexandria, those same rays strike at a slight angle, and the post casts a shadow.

Eratosthenes understood the key idea:

That meant that if he measured this small angle between the two cities, he could determine what fraction of the Earth’s entire circumference the distance between them represented.

And then—he could measure the whole Earth.

A Small Angle and an Enormous Planet

Eratosthenes measured the shadow in Alexandria and determined the angle.

It was about 7.2°.

At first glance, that seems very small.

But a full circle contains 360°.

Now comes the simple calculation:

So the distance from Alexandria to Syene represented approximately one-fiftieth of the Earth’s circumference.

Only one more thing was needed: the distance between the two cities.

It was estimated at about 5,000 stadia. The stadion was an ancient unit of length, and its exact value varied in different places and periods.

Now everything came down to a single multiplication:

That was how Eratosthenes obtained his estimate of the Earth’s circumference.

Today it is difficult to convert his result into kilometers with complete certainty, because different lengths of the stadion were used in antiquity. But even without an exact conversion, the achievement is astonishing:

more than two thousand years ago, a human being arrived at a remarkably close estimate of the size of our planet.

Only the Sun, a Shadow, and the Human Mind

Think about what Eratosthenes had actually done.

He had not sailed around the Earth.

He had never seen it from space.

He could not photograph its curvature.

He simply observed the shadow cast by an ordinary vertical post.

Then he brought together a few simple things:

The Sun.
A shadow.
An angle.
The distance between two cities.
Mathematics.

And a small shadow in Alexandria allowed the human mind to reach around an entire planet.

Who Was Eratosthenes?

Measuring the Earth was only one of his achievements.

Eratosthenes was one of the most versatile scholars of the ancient world. He worked in mathematics and astronomy, geography and history, literature and poetry.

His name is closely associated with the rise of scientific geography—the attempt to describe the Earth not only through travelers’ stories, but through measurements, mathematical calculations, and maps.

He created a map of the world known to him and used lines that foreshadowed the later system of parallels and meridians.

In mathematics, his name lives on in the Sieve of Eratosthenes—a simple and elegant method for finding prime numbers that children still learn today.

He even tried to construct a unified chronology of the past, comparing ancient events, generations, and lists of kings.

Most of his writings have not survived.

But one of his ideas has endured for more than two thousand years:

sometimes, to answer an enormous question, you need only look very carefully at something very small.

For Eratosthenes, that clue was a shadow.

Now Look at the Postcard Again

Why does the post in Syene cast almost no shadow, while the one in Alexandria does?

Look at the rays of the Sun and the curve of the Earth.

That tiny angle is what allowed Eratosthenes to measure an entire world.