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Chapter 1: Problem 4

How did Eratosthenes calculate the Earth's circumference?

Short Answer

Expert verified
Eratosthenes calculated the Earth's circumference by measuring the angle of the Sun's rays at noon during the summer solstice in two different cities, Alexandria and Syene. With a 7.2-degree angle difference and an estimated distance of 5,000 stadia between the cities, he used proportion, assuming Earth as a sphere, to calculate the circumference as \(C \approx 250,000\) stadia or about 39,375 kilometers, which is reasonably close to the actual value of 40,075 kilometers.

Step by step solution

01

Select measurement locations

Eratosthenes chose two cities in Egypt for his measurements: Alexandria and Syene (now Aswan). These cities happen to lie approximately on the same meridian, which makes it easier to compare the angle of the Sun's rays. Syene is located almost exactly south of Alexandria.
02

Measure the angle of the Sun's rays at noon

Eratosthenes made his measurements on the day of the summer solstice, when the Sun is directly overhead at noon at the Tropic of Cancer. In Syene, at that time, the Sun was directly overhead (at the zenith), so the angle of the Sun's rays was 0 degrees from the vertical. In Alexandria, however, he observed that the angle of the Sun's rays formed a 7.2-degree angle with the vertical, as they were not directly overhead.
03

Determine the distance between the cities

Eratosthenes needed to know the distance between Alexandria and Syene to calculate the Earth's circumference. He used the standard distance measurement of his time, called a stadia. He estimated the distance between the two cities to be about 5,000 stadia. Nowadays, we know that this distance is approximately 800 kilometers.
04

Consider Earth as a sphere

Eratosthenes assumed that the Earth was a perfect sphere. Based on this assumption, he also assumed that the angle between two cities along the Earth's surface was equal to the angle difference in the Sun's rays observed at the two cities. In other words, the angle formed by the rays in Alexandria and Syene, which was 7.2 degrees, corresponded to the angle formed by two lines drawn from the Earth's center to the cities.
05

Calculate Earth's circumference

Since we're assuming Earth as a sphere, we know that its surface is 360 degrees. So we can set up a proportion like this: \[\frac{7.2^\circ}{360^\circ} = \frac{5,000\,\text{stadia}}{C}\] Where \(C\) is the Earth's circumference in stadia. To find the value of \(C\), we can cross-multiply and then divide by 7.2, like this: \[C = \frac{360^\circ \cdot 5,000\, \text{stadia}}{7.2^\circ}\] \[C \approx 250,000\, \text{stadia}\] So, Eratosthenes calculated the Earth's circumference to be about 250,000 stadia. In modern units, this would be approximately 39,375 kilometers. The actual value of Earth's circumference is around 40,075 kilometers, which shows how accurate Eratosthenes' method was for his time.

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