Discover how the Greek mathematician, Eratosthenes, measured the Earth using just a stick on the summer solstice.
June 21st is the summer solstice, the day that marks the beginning of summer.
Over 2,000 years ago, a mathematician named Eratosthenes had the brilliant idea of calculating the Earth’s radius with just a stick. How did he do it? Let’s look!!
Procedure
- The first thing he did was obtain a stick (a very important step if you want to calculate the Earth’s radius with a stick, as redundant as it may sound).
- At solar noon on the summer solstice (exactly), he placed the stick at a 90-degree angle to the ground.
- He measured the angle formed by the top of the stick and the end of its shadow.
4. Once he had the angle measurement, he needed one more crucial piece: the distance from where he was (Alexandria) to a place on the Tropic of Cancer** (he used Siena). To calculate this distance, he measured the length of a slave’s stride and had him walk the entire distance from Alexandria to Siena.
5. Once he had all the data, he made the following calculation to find the Earth’s radius (important: the angle must be in radians):
r = distance / angle
And that’s how Eratosthenes calculated the Earth’s radius with this simple reasoning, or was it really that simple? Let’s dive into the explanation!!
Explanation of How Eratosthenes Measured the Earth with a Simple Stick
At solar noon on the summer solstice, the Sun’s rays fall on the Tropic of Cancer in such a way that no object casts a shadow.
If you stand anywhere at that hour and measure the angle formed by the stick with the shadow it casts, you’ll notice that this angle is the same as the arc formed from where you’re standing to the Tropic of Cancer.
Demonstration
Of course, it’s not much use just telling you all this without providing a demonstration—that’s what we’re here for! Now, let me show you step by step how I replicated what Eratosthenes did more than two thousand years ago.
- The first thing I did was take a stick. Since I couldn’t find one, I decided to use a bottle (you know, got to recycle).
- I waited for solar noon on June 21st, 2022 (14:20), and when the time arrived, I placed the bottle on a flat surface. I measured the length of the bottle and the shadow it cast, and the data was as follows:
- Bottle length = 26 cm
- Shadow length = 10 cm
- Once at home, I calculated the tangent arc of these two measurements by dividing them to find the angle in radians:
arctan(10/26) = arctan(0.38) = 0.3631 rad
- Finally, I divided the distance from where I was to the Tropic of Cancer (which was 2315 km):
r = 2315 / 0.3631 = 6375 km
