# How Eratosthenes Calculated the Circumference of the Earth Two Millennia Ago

An ancient foray into scientific calculation with sticks, sunlight, and ingenuity

Eratosthenes, a Greek polymath and the director of the Library of Alexandria, lived in the 3rd century BCE. Equipped with only sticks, his eyes, his feet, and his brain, he both proved that the Earth was round and deduced the circumference of the Earth with a <1% margin of error (1).

One assumption and two observations were key to Erathosthenes’ calculations. He first assumed that the Sun is very far from the Earth, which would mean that the Sun’s rays are almost parallel when they hit Earth (2). His first observation was actually a report from the Egyptian city of Syene, where, at noon on June 21st (the date of the summer solstice), vertical objects did not cast a shadow––an effect that only occurs when the sun’s rays are directly perpendicular to the Earth’s surface (3). He later observed that, on the same date at the same time, a stick planted directly in the ground did indeed cast a shadow––a shadow that measured about 7.2 degrees. These observations are shown in Figure 1.

These facts led to the conclusion that the Earth must be round; if the Earth was flat, then the Sun’s rays would be perpendicular to Earth in Alexandria and objects in the city would not cast a shadow.

Erathosthenes then wondered what the circumference of Earth was. Using the fact that alternate interior angles of parallel lines are equal (and you thought geometry class was useless), Erathosthenes realized that the arc of the Earth between Syene and Alexandria must also be 7.2 degrees (2). There are 360 degrees in a circle, and 3607.2 = 50, so the circumference of the Earth must be 50 times the distance between the two cities. If Erathosthenes could find the distance between the two cities, then he would be able to calculate the circumference of the Earth. And that’s exactly what he did; Erathosthenes paid someone to physically walk and count the number of steps between the two cities, eventually concluding that the distance between Syene and Alexandria was about 800 km (1). Thus, he concluded that the earth was around 50 * 800 km = 40,000 km in circumference. Today, with satellite observations, the Earth’s circumference is known to be 40,075 km.

Exploring the history of science is worthwhile. Erathosthenes’ approach teaches us valuable lessons in observation, and the importance of investigating strange phenomena. His ingenuity inspires us to tackle great challenges, despite the technological limitations of our time.

Bibliography – APA

1. Encyclopædia Britannica, inc. (n.d.). Eratosthenes. Encyclopædia Britannica. Retrieved September 24, 2022, from https://www.britannica.com/biography/Eratosthenes
2. Business Insider. (n.d.). How the ancient greeks proved Earth was round over 2,000 years ago. Business Insider. Retrieved September 24, 2022, from https://www.businessinsider.com/how-greek-eratosthenes-calculated-earth-circumference-2016-6
3. CarlSaganPortal. (2009, April 24). Carl Sagan – Cosmos – Eratosthenes. YouTube. Retrieved September 24, 2022, from https://www.youtube.com/watch?v=G8cbIWMv0rI

Images

Figure 1: A diagram summarizing Erathosthenes assumptions and observations

https://www.vanderbilt.edu/AnS/physics/astrocourses/ast203/eratosthenes.html

Lahaina Noon – The phrase Hawaiians use to describe the phenomenon when vertical objects do not have a shadow