Real Mathematics – Strange Worlds #4

Orange Season

I like using oranges for mathematics because I think they taste awesome. Also, its shape is close to Earth’s shape which I believe is a cool resemblance. Even though I’d love to, I can’t take credit for using orange with examples about Earth. It belongs to a deep and important dispute in the history of science.

17th century is known as the century when the modern science was born. Giovanni Cassini, an Italian astronomer was born in this century along with so many other important figures. He is famous with discovering Saturn’s rings and its four big moons. In addition, he went into a big dispute with Isaac Newton, who is known as the father of modern physics. I’ll save Newton’s introduction for another article.

Dispute began with Cassini’s measurements which he did with his son. Those measurements led him to a wrong conclusion. He resulted that Earth is elongated at the poles, like a lemon. On the other hand, Newton had explained that Earth is flattened at the poles, like an orange. This was a result Newton reached after his gravitational laws. Unfortunately for Newton, Cassini openly rejected Newton’s gravitation laws which made him adopt his stance even harder after the measurements he had for determining the shape of Earth.

gravity-orange-lemon-small (2)
Newton suggested that Earth’s shape is like an orange as Cassini wrongly claimed that it looks like a lemon.

This dispute continued for almost forty years until French Geodesy Mission (1736-1744) was completed. Measurements and calculations had proven Newton right: Earth was shaped like an orange.

Finding Circumference

Stick two equal-sized straws on an orange.


Direct a flash lamp to one of the straws so that it doesn’t have a shadow.


If the distance between those straws is given, is it possible to calculate the circumference of the orange?

Second-Best of Everything

Eratosthenes was born in the ancient city of Cyrene, at around 3rd century BC. He is known as the person who discovered (or invented) geography. There is an interesting take from Thomas Hearth (1861-1940) who was an important historian of mathematics specifically on ancient Greek mathematics: “Eratosthenes was great on every subject of science, but he was never ‘the best’ in one area. You may imagine him like an athlete who competes in every branch of Olympics but comes second in those competitions.”

Even though he founded an important science branch like geography, Eratosthenes is known with calculating the circumference of the Earth almost 2200 years ago. And he did it with a surprising accuracy. But how did he manage it?

Eratosthenes realized on a summer day that he had no shadow at noon in Cyrene. Although at the same time when he tried it in Alexandria he saw a shadow. Eratosthenes believed that Earth was round and Sun is too far away from Earth. He also thought of light rays travel as parallel lines which is why he made an experiment that took place in Cyrene and Alexandria. He observed a tall tower’s shadow in Alexandria when there is no shadow in Cyrene. Eratosthenes measured that the shadow of the tower makes a 7,2-degree angle with its bottom. He also had the distance between two cities measured that enabled him to calculate the circumference of the Earth.

A as Cyrene and B as Alexandria.

Eratosthenes used an ancient Greek measurement called “stadium” in his calculation. Since a stadium means something between 154- 215 meters, we are not 100% sure what his calculation was. In the end, his solution and method was good enough to put him into the highlights of the history of science.

Circumference of the Orange

Now that you know Eratosthenes’ method, could you find the circumference of the orange?

Hint: The angle straw makes is 12 degrees and distance between straws is exactly 1 cm.

M. Serkan Kalaycıoğlu

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