Question

How did Eratosthenes calculate the Earth's circumference? (A)

Short answer

Eratosthenes calculated the Earth's circumference by selecting two cities, Alexandria and Syene, and measuring the angle of the sun's rays at each location on the summer solstice. He found an angular difference of 7.2 degrees between the two cities, which corresponds to 1/50 of the Earth's total circumference. Knowing the distance between the cities was about 800 kilometers, he multiplied this value by 50 to estimate the Earth's circumference to be around 40,000 kilometers, a value close to the actual measurement.

Step-by-step solution

Choose two locations

Eratosthenes selected two cities: Alexandria and Syene (now known as Aswan) in Egypt. He knew the approximate distance between them, which was about 800 kilometers (500 miles).

Measure the angle of the sun's rays at each location

On the day of the summer solstice, when the sun was at its highest point in the sky, Eratosthenes measured the angle of the sun's rays in each city. In Syene, the sun was directly overhead, casting no shadow, so the angle was 0 degrees. In Alexandria, he measured the angle to be about 7.2 degrees.

Calculate the angular difference between the two cities

Eratosthenes found the difference between the angles of the sun's rays in both cities, which was 7.2 degrees.

Convert the angular difference to a fraction of the Earth's total circumference

Since there are 360 degrees in a circle, Eratosthenes determined the fraction of the Earth's total circumference by dividing the angular difference (7.2) by 360. This yielded 7.2 / 360 = 1/50.

Calculate the Earth's circumference

Eratosthenes knew the distance between Alexandria and Syene to be 800 kilometers (500 miles). He multiplied this distance by 50 (since 1/50 was the fraction of the Earth's total circumference represented by the angular difference) to estimate the Earth's circumference. So, 800 km x 50 = 40,000 km (25,000 miles). Based on this calculation, Eratosthenes found that the Earth's circumference was approximately 40,000 kilometers (25,000 miles), which is quite close to the actual value of about 40,074 kilometers (24,901 miles).

Key concepts

Ancient Greek Science

The realm of ancient Greek science was filled with discoveries and innovation, ripe from the rational thinking and curiosity of ancient Greek scholars. Among the foremost pioneers, Eratosthenes stands out. Living in the third century BCE, Eratosthenes was not just a scientist but also a poet, librarian, and mathematician. He lived during a period known for substantial intellectual achievements.
Eratosthenes' approach represented the empirical side of Greek science, which emphasized observations and measurements. He innovatively used straightforward observations to deduce the Earth's size, similar to how ancient Greeks sought logical explanations for natural phenomena.

• Empirical Method: Emphasis on observation and measurement.
• Interdisciplinary Approaches: Eratosthenes' work showcased the integration of geography, mathematics, and astronomy.

He embodied the spirit of inquiry that defined ancient Greek science, showing how diverse knowledge fields can combine to solve complex problems.

Geodesy

Geodesy, the science of measuring and understanding Earth's geometric shape, gravitation, and spatial orientation, was at the core of Eratosthenes' circumferential calculation. Eratosthenes, who was primarily a geographer, applied his knowledge of the Earth's layout to tackle celestial measurements. His experiment relied on the assumption that Syene and Alexandria sat directly north-south of one another, within generally accepted distances.
With these cities in mind, he was able to precisely measure surface distances, transferring results into significant insights about the whole of Earth.

• Surface Measurements: Understanding the terrain and distances between specific earthly locations.
• Assumptions of Uniformity: Assuming a spherical Earth, he linked surface measurements to larger dimensions.

Eratosthenes' method was revolutionary because it blended practical geographic surveys with spherical geometry, a testimony to the integration of geography with mathematical theory.

Solar Angle Measurement

Central to Eratosthenes’ calculation was the measurement of solar angles, a pivotal part of mathematical astronomy. This involved using the angle of the sun’s rays relative to the ground at two different locations to derive important information about the Earth’s circumference. Eratosthenes cleverly used the fact that on the summer solstice at noon, the sun was directly overhead in Syene, casting no shadow at all, while in Alexandria, it made a specific angle.
By measuring the angle of 7.2 degrees in Alexandria, he calculated the angular distance on Earth that corresponds to this solar angle.

• Angle of Elevation: Measuring the height of the sun can indicate horizontal distances.
• Basic Instruments: Simple tools like gnomons or sundials sufficed for these observations.

These insights linked sun positions with terrestrial measures, a fundamental technique in solar angle analysis and instrumental in historical astronomical practices.

Mathematical Astronomy

Eratosthenes' method for calculating the Earth's circumference was a brilliant blend of geometry and astronomy, embodying mathematical astronomy. This science, which combines mathematical principles with astronomical phenomena, allowed Eratosthenes to transform simple observations into quantitative information. In this realm, the 360-degree division of circles was crucial.
By recognizing that 7.2 degrees made up the arc between Alexandria and Syene, he could extrapolate the length of a circle's entire circumference by basic proportion.

• Proportionality: The relationship between arc sections and full circles.
• Geometry and Spheres: Relating flat geometry to round surfaces.

Eratosthenes' accuracy evidenced how effectively ancient scholars applied mathematical frameworks to cosmic questions, capturing the true spirit of early astronomical endeavors.