Eratosthenes calculated the Earth’s circumference more than 2,200 years ago [Here’s how]

In an era long before satellites and advanced technology, an ancient Greek mathematician named Eratosthenes accomplished an extraordinary feat – he calculated the Earth’s circumference with remarkable accuracy, even without leaving Egypt. Armed only with a stick, the angle of a shadow, and the distance between two cities, Eratosthenes was able to deduce that our planet was a sphere and estimate its size. This achievement in 240 BC, more than 2,200 years ago was not only a testament to human ingenuity but also set the groundwork for future geographical and astronomical studies. His approach was brilliantly simple, yet so effective that it came incredibly close to the measurements we use today, over two millennia later.

Eratosthenes’ rudimentary yet ingenious technique laid the foundations for later scientific discoveries and even today stands as a remarkable feat of reasoning. Armed with the scientific curiosity that defined his career, Eratosthenes made use of straightforward observations and simple geometry to conclude what would be one of the most fascinating truths about our planet: the measurement of its circumference. His method offers insights into how basic principles can unveil complex realities.

Despite claimed by many reputable sources, the idea of Earth being as smooth as a billiard ball is a myth.

Related: No, the Earth is NOT As Smooth As a Billiard Ball! [Here’s Why]

Eratosthenes’ method to calculate the Earth’s circumference

Eratosthenes, who was the chief librarian at the Library of Alexandria around 240 BC, used a pretty straightforward way to figure out the Earth’s size. Here’s how he did it, step by step:

  1. Two Cities, One Observation: He focused on two cities: Syene (modern Aswan, Egypt) and Alexandria. He knew that during the summer solstice, at noon, the Sun was directly overhead in Syene.
  2. Subsolar Point: The term “Subsolar Point” is where the Sun is directly overhead at noon. For Syene, this was the case during the summer solstice. When the Sun is directly overhead, you’ll see that deep wells have no shadow at the bottom, because the Sunlight can reach all the way down. This point, where the Sun is directly overhead, is important because it serves as a reference. Syene is slightly north of the Tropic of Cancer. On the summer solstice at local noon on the Tropic of Cancer, the Sun would appear at the zenith, directly overhead (sun elevation of 90°).
  3. Measuring Shadows in Alexandria: On the same day and time, Eratosthenes measured the angle of the shadow cast by a stick in Alexandria. He found this angle to be about 7.2 degrees. This was the difference between the Sun being directly overhead (at the subsolar point in Syene) and its position in Alexandria.
  4. Calculating the Earth’s Slice: Now, here comes the clever part. A full circle has 360 degrees. If Alexandria and Syene are 7.2 degrees apart in terms of the Sun’s position, then they are 7.2/360 = 1/50 of the Earth’s full circle apart.
  5. Distance Between Cities: Eratosthenes knew that the distance between Syene and Alexandria was about 800 km (he used a unit called “stadia”, but it roughly converts to this distance in modern units). Caravan travelers knew the distance between the cities to be about 5,000 stadia. Carl Sagan even says that Eratosthenes paid a man to walk and measure the distance.
  6. Finding the Full Circumference: To calculate the Earth’s full circumference, you have to multiply the distance between the two cities by 50 (because 1/50th of a circle corresponds to 800 km between the cities). So, 800 km multiplied by 50 gives you 40,000 km.
  7. Final Result: Eratosthenes concluded that the Earth’s circumference was around 40,000 km, which is impressively close to the modern measurement of about 40,075 km.

Why is the Subsolar Point Important?

The subsolar point serves as a ‘benchmark’ or ‘starting point.’ Knowing that the Sun was directly overhead in Syene meant that there was a reference point to measure how “off” the Sun’s rays were in Alexandria. This difference, measured in degrees, helped him figure out the slice of Earth between the two cities, which in turn led him to the Earth’s full circumference.

An illustration depicting how, over 2,200 years ago, Eratosthenes managed to estimate the Earth circumference
An illustration depicting how, over 2,200 years ago, Eratosthenes managed to estimate the Earth’s circumference without ever leaving Egypt. He observed that on the summer solstice at local noon in Syene (modern-day Aswan), the Sun was positioned directly overhead. This was evident as no reflection of the Sun could be seen in the water when someone peered into a deep well at that moment, due to the shadow obscuring the Sun’s reflection. In Alexandria, on the same day and time, he assessed the Sun’s angle by observing the shadow cast by a vertical stick. By constructing a scale diagram of this scenario, which formed a right-angled triangle with the stick and its shadow, he found the angle to be equivalent to 1/50th of a full circle. Assuming the Earth was spherical and knowing the distance and direction to Syene, he deduced that the Earth’s circumference must be fifty times the distance between Alexandria and Syene.

So, if one would know the distance between Syene and Alexandria, s/he could calculate the circumference of the Earth. 360/7 is close to 1/50th of a circle, so Eratosthenes concluded that the Earth’s circumference was fifty times that distance.

C = 360/7 x d

(C: circumference of Earth, d: the distance between Syene and Alexandria)

Today, we know that Earth’s equatorial circumference is 40,075.017 km (24,901.461 miles).

The stadion (Latin for stadium, with the plural form being stadia) served as a unit of distance frequently employed in ancient times. However, the exact length of a stadion varied significantly depending on the region and era.

It’s suggested by some that Eratosthenes might have used the Olympic stadion, measuring 176.4 meters (approximately 579 feet), for his calculations. This assumption would lead to an estimated Earth’s circumference of 44,100 kilometers, presenting an error margin of about 10%. Nevertheless, the Italian stadion, measuring 184.8 meters (around 606 feet), later gained widespread acceptance as the stadion measurement about 300 years after Eratosthenes’ time.

Using this measurement for the stadion, the calculated circumference of the Earth would be 46,100 kilometers, indicating an error margin of 15%. Despite the discrepancies, these figures are remarkably accurate for their time.

Please note that Eratosthenes made two important assumptions, neither of which is perfectly accurate:

  1. The distance between Alexandria and Syene was 5,000 stadia,
  2. That the Earth was a perfect sphere.

1,700 years after Eratosthenes’ death, while Christopher Columbus studied what Eratosthenes had written about the size of the Earth, he chose to believe, based on a map by the Italian astronomer, mathematician, and cosmographer Paolo dal Pozzo Toscanelli (1397 – 10 May 1482), that the Earth’s circumference was one-third smaller. Had Columbus set sail knowing that Eratosthenes’ larger circumference value was more accurate, he would have known that the place where he made landfall was not Asia, but rather a New World.

In the video below, which is an excerpt from Carl Sagan’s “Cosmos: A Personal Voyage” (a thirteen-part television series written by Carl Sagan, Ann Druyan, and Steven Soter, with Sagan as the presenter), Sagan explains how Eratosthenes calculated Earth’s circumference in the 3rd century BCE.

Cosmos – Eratosthenes calculates Earth’s circumference. In the 3rd century BCE, Eratosthenes calculated the circumference of the Earth to high precision.

Sources

M. Özgür Nevres
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