In the annals of scientific history, few names stand out as prominently as Eratosthenes. A polymath of the ancient world, Eratosthenes made significant contributions to various fields, including geography, mathematics, and astronomy. However, he is best known for his remarkable method to estimate the circumference of the Earth, a feat that has earned him the title of the “Father of Geography.”
Who Was Eratosthenes?
Eratosthenes was born in 276 BCE in Cyrene, a Greek city in modern-day Libya. He was a man of many talents and interests, which led him to study in various fields. He was a poet, an astronomer, a geographer, and a mathematician. His intellectual prowess caught the attention of the Ptolemaic rulers of Egypt, and he was invited to Alexandria to serve as the chief librarian of the Great Library of Alexandria, one of the most significant centres of learning in the ancient world.
Eratosthenes’ Method to Estimate the Earth’s Circumference
Eratosthenes’ most famous achievement is his calculation of the Earth’s circumference. His method was a brilliant combination of observation, geometry, and logical reasoning. Here’s a step-by-step breakdown of his approach:
Observation of the Sun’s Position: Eratosthenes noticed that on the summer solstice, the Sun was directly overhead in the city of Syene (modern-day Aswan, Egypt). This meant that the Sun’s rays were shining straight down a deep well, casting no shadow. However, at the same time in Alexandria, which is north of Syene, the Sun was not directly overhead. Instead, it cast a shadow, indicating that the Sun’s rays were coming in at an angle. (Note that the reason Eratosthenes chose Alexandria, is that he assumed Alexandria was located on the same meridian as Syene. This is a key criteria for this method to work. Though in reality this is not quite the case, which causes some slight inaccuracy in Eratosthenes estimate of the Earth circumference.)
Measurement of the Shadow’s Angle: In Alexandria, Eratosthenes measured the angle of the shadow cast by a vertical stick (a gnomon, the raised arm/stick of a sundial) at noon, on the summer solstice. He found that the angle was about 7.2 degrees, which is approximately 1/50th of a full circle (360 degrees).
Assumption of a Spherical Earth: Eratosthenes assumed that the Earth was a sphere, a concept that was widely accepted among educated Greeks at the time. He also assumed that the Sun’s rays were parallel, which is a reasonable approximation given the vast distance between the Earth and the Sun.
Calculation of the Earth’s Circumference: Using the information he had gathered, Eratosthenes reasoned that the distance between Alexandria and Syene was about 1/50th of the Earth’s circumference. He knew the distance between the two cities was approximately 5,000 stadia (a Greek unit of measurement: the exact length of a stadion is uncertain, but it is generally believed to be around 157.5 meters). So the distance between Syene and Alexandria is approximately 787.5km. By multiplying this distance by 50 to estimate the Earth’s circumference.
5,000 stadia * 50 = 250,000 stadia (or 787.5km * 50 = 39,375km)
This would make Eratosthenes’ estimate of the Earth’s circumference approximately 39,375 kilometres, which is remarkably close to the modern value of 40,075 kilometres.
Eratosthenes’ method to estimate the Earth’s circumference is a testament to the power of observation, reasoning, and mathematical prowess. His achievement is all the more remarkable considering the limited technology and resources available to him.
Python Challenge
Let’s write a small Python script to enable us to apply this method with different angle measurements made on different location on planet Earth, and let us see if we get a similar estimation of the Earth’s circumference.
Our Python program will need to:
-
Take two inputs: the angle of the shadow and the distance in km from where the angle was measured and Syene’s well
Use a mathematical formula based on Eratosthenes method:
Output the estimated circumference of planet Earth in km, rounded to 3 decimal places
Calculate the percentage of accuracy of this estimate, rounded to 2 decimal places, knowing that we now know that the circumference of planet Earth is 40,075km. The formula to claculate the percentage of accuracy is as follows:
Python Code
Type your code below:
Testing
To test your algorithm we have collected the approximate distances from Syene (Aswan, Egypt) of some different locations, all on the same meridian as Syene. For each location, we have recorded the angle of the shadow cast by the Sun at noon on the summer solstice (June 21st). Here is our test data:
| City | Distance from Syene (km) | Angle of the Shadow (degrees) |
|---|---|---|
| Location 1: Alexandria, Egypt | 800 | 7.2 |
| Location 2: | 850 | 7.8 |
| Location 3: | 2,000 | 18.2 |
| Location 4: | 2,500 | 22.8 |
| Location 5: | 2,800 | 25.6 |
| Location 6: | 3,500 | 32.0 |
| Location 7: | 4,000 | 36.6 |
| Location 8 | 9,000 | 82.2 |
Can you check that your algorithm gives accurate estimate of the Earth circumference for each set of data provided in the table above.
Solution...
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