Question

How did Eratosthenes use Sun angles to figure out that the 5000 -stadia distance between Alexandria and Syene was $1/50$ of Earth's circumference? Once he knew this fraction of Earth's circumference, how did he calculate the entirety of Earth's circumference?

Short answer

Eratosthenes used geometric principles and the angle of the sun at the summer solstice to deduce that the Earth's circumference was 250,000 stadia.

Step-by-step solution

Identify the Method Used

Eratosthenes used geometric principles to deduce the size of the Earth. He knew that at noon on the summer solstice, the sun was directly overhead in the city of Syene (casting no shadow), but at the same time in Alexandria, the sun was at an angle of approximately 7.2 degrees from the zenith, casting a shadow. These places were known to be roughly 5000 stadia apart.

Constructing Similar Triangles

Next, two right-angled triangles can be constructed, with one vertex at the center of the Earth, and the long side from the center of the Earth to the respective town (Syene and Alexandria). The angle at the peak of both right triangles is equivalent to the angle that sunlight makes with the vertical direction at the given town. Since one angle of two right-angled triangle is the same, they are similar triangles, meaning that the ratios of their corresponding sides are identical.

Determine Earth's Fraction with Respect to Stadia

On the summer solstice, the precise calculation of the sun's angle at noon in Alexandria and Syene would yield an angle of around 7.2 degrees, which is $1/50$ of a full circle. Therefore, the distance between Syene and Alexandria, 5000 stadia, is $1/50$ of the Earth's circumference.

Calculating the entire Circumference of the Earth

After he had calculated this fraction of the Earth's circumference, Eratosthenes could easily compute the entirety of Earth's circumference by multiplying by the reciprocal of the fraction. The total circumference of the Earth is then calculated as the product of 50 and the measured distance in stadia, which comes to 250,000 stadia.

Key concepts

Eratosthenes

Eratosthenes was a Greek mathematician and astronomer who lived around 276–194 BCE. He is famously known for his early and remarkably accurate calculation of Earth's circumference.
Eratosthenes' brilliance lay in his application of known geometry and observational techniques to solve real-world problems. He took advantage of his knowledge about the behavior of the sun at different latitudes to deduce Earth's size, an incredible feat considering the limited tools and information available at the time.

• He observed the sun's direct overhead phenomenon in Syene (modern-day Aswan) at noon during the summer solstice.
• Simultaneously, he noted the sun cast a shadow at an angle of 7.2 degrees in Alexandria, located north of Syene.

This insight forms the basis of his method, demonstrating the power of observation combined with scientific reasoning.

Sun Angle Method

The Sun Angle Method involves measuring angles of the sun's rays at two different locations to determine the curvature of Earth. Eratosthenes applied this method by focusing on two key facts:
1. **No Shadow in Syene:** In Syene, the sun was directly overhead at noon, meaning no shadow was cast by vertical objects.
2. **Shadow in Alexandria:** In Alexandria, the sun formed an angle of 7.2 degrees with the zenith at the same time.
This discrepancy suggested that Earth is curved, and by measuring this angle and knowing the physical distance between the two cities, Eratosthenes could make calculations about Earth's size.

• This angle provided the needed information about Earth's curvature.
• The consistent shadow angle allowed for the deduction of distances correctly related to Earth's surface curvature.

Through these observations, Eratosthenes calculated that the 5000 stadia between the two cities represented 1/50th of Earth's circumference.

Geometric Principles

Geometric principles served as the backbone for Eratosthenes' calculations. These principles, which include measurements of angles and distances, were pivotal in linking the sun's angle to Earth's curve.
Eratosthenes used simple geometry to translate local measurements into global implications:

• He worked with the concept of a circle, where a full circle is 360 degrees.
• The sun's 7.2-degree angle represented a fractional arc of Earth's total circumference.

By harnessing these basic yet powerful principles of geometry, he elaborated how a small angle of shadow equated to a large section of Earth. This method assumed Earth was a sphere, allowing him to accurately deduce the scale of the entire planet based on local observations.

Similar Triangles

The principle of similar triangles was a key mathematical insight for Eratosthenes in his calculation.
Eratosthenes observed that two right triangles could be constructed:

• One at Syene where the sun was directly overhead, hence an angle of 0 degrees from the zenith.
• Another at Alexandria with a 7.2-degree angle from the zenith.

The similar triangles' concept states that if two angles of one triangle are equal to two angles of another triangle, the two triangles are similar. This means the ratios of their corresponding sides are identical.
By applying this principle to the 7.2-degree angle at Alexandria, Eratosthenes was able to infer that the arc distance between Syene and Alexandria represented the same fraction of a full circle as the angle did. The use of similar triangles allowed him to extend his observations on a local level to Earth's entire size, forming a vital part of his calculation.