Eratosthenes of Cyrene. Calculation of the radius of the Earth.

Date: 10/01/2024 Author: Antonio H.

Eratosthenes of Cyrene (276 BC - 194 BC). The radius of the Earth

Eratosthenes was a mathematician, astronomer and geographer, born in 276 B.C. in Cyrene, an ancient Greek city in present-day Libya, known, among other contributions, for his work on estimating the radius of the earth.

As director of the Library of Alexandria (Egypt) he was aware of observations indicating that in Syene, now Aswan, at noon on the summer solstice (when the Sun reaches its highest position in the sky) it was observed that objects cast no shadow and the light entered the wells to the very depths. This meant that the sun's rays fell perpendicular to the earth in that area.

In Alexandria, on that same day, the objects did cast shadows. That is, those same sunbeams did not fall perpendicularly, but at a certain angle.

It should be noted that Eratosthenes already held that the earth was curved, so this difference in the angle of incidence of the sun's rays reinforced his belief.

He also believed that the sun was far enough away to consider that its rays would reach the earth practically parallel to each other.

Moreover, if the sun entered the well all the way to the bottom, he could infer that, if they were prolonged, they would reach the center of the earth.

The distance between Alexandria and Syene was about 5,000 stades, a measure used at the time which, although not known exactly, is estimated to be equivalent to about 158 meters.

Eratosthenes was able to measure the angle at which the sun's rays hit Alexandria and the result was approximately 7º 12'. The method he used to obtain it is not known. .

Eratóstenes Alejandría Siena

Knowing that the full circumference comprises 360º, Eratosthenes divided by the 7.2º measured, resulting in approximately 50 times the circumference:

360 / 7.12 = 50

from which he inferred that the circumference of the earth should be about 50 times the distance between Alexandria and Siena, that is 50 x 5000 = 250,000 stades.

If we take into account the estimated value in meters for a stade we will have that the circumference of the earth would be:

250.000 stades x 158 meters/stade = 39.500.000 meters = 39.500 km

By that time Archimedes had already made an approximate calculation for the value of PI by setting it at 22/7 and the relationship between the length of a circle and its radius was known:

Circ. length formula

Clearing the radius in the above formula:

Circ. radio formula

Using the values obtained by Eratosthenes:

Radio Eratóstenes

If we take as valid the 158 meters that each stade was supposed to measure:

39.772 · 158 = 6.283.976 meters = 6.284 km.

The Earth is not perfectly spherical, so it does not make sense to speak of a single radius, we can find that the radius at the equator is 6,378 km while the polar radius is 6,357 km. As a reference, 6,371 km is considered to be the average radius in a spherical modeling of the Earth.

This is an example of the application of basic geometry and that ancient civilizations were not necessarily intellectually primitive.

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