Eratosthenes: Circumference of Earth Calculated with a Stick!

Perhaps one of the most astounding achievements of the human mind occurred for a Greek scientist in Alexandria named Eratosthenes who figured out how to measure the size of the Earth, about 2,200 ago. His estimation came within about two percent of the correct answer!

What he did was simple and genius, just by working with a stick!
He noted the angle of the shadow cast by the sun on a certain day, in Alexandria… But he also knew, that in a water-well several hundred miles away, a stick cast no shadow on noon that day, because of the angle of the sun. Having sent out a scout to measure the distance between Alexandria and the well in Syene to be about 5,000 stadia (or 500 miles or 800 kilometers), he applied a little primitive trigonometry, to determine how much of an arc on a circle separated these two points being about 7.5 degrees. Then, applying a little bit of algebra, he quickly calculated the size of the Earth to be about 40,000 km with very good accuracy.

 Since the spherical shape was the most widely supported during the Greek Era, efforts to determine its size followed. Plato determined the circumference of the earth to be 40,000 miles while Archimedes estimated 30,000 miles. Plato’s figure was a guess and Archimedes’ a more conservative approximation. Meanwhile, in Egypt, a Greek scholar and philosopher, Eratosthenes, set out to make more explicit measurements.

He had observed that on the day of the summer solstice, the midday sun shone to the bottom of a well in the town of Syene (Aswan). At the same time, he observed the sun was not directly overhead at Alexandria; instead, it cast a shadow with the vertical equal to 1/50th of a circle (7° 12′). To these observations, Eratosthenes applied certain “known” facts (1) that on the day of the summer solstice, the midday sun was directly over the line of the summer Tropic Zone (Tropic of Cancer)-Syene was therefore concluded to be on this line; (2) the linear distance between Alexandria and Syene was 500 miles; (3) Alexandria and Syene lay on a direct north-south line.

ERATOSTHENES’ METHOD FOR DETERMINING THE SIZE OF THE EARTH

From these observations and “known” facts, Eratosthenes concluded that, since the angular deviation of the sun from the vertical at Alexandria was also the angle of the subtended arc, the linear distance between Alexandria and Syene was 1/50 of the circumference of the earth or 50 x 500 = 25,000 miles. A currently accepted value for the earth’s circumference at the Equator is 24,901 miles, based upon the equatorial radius of the World Geodetic System. The actual unit of measure used by Eratosthenes was called the “stadia.” No one knows for sure what the stadia that he used is in today’s units. The measurements given above in miles were derived using one stadia equal to one-tenth statute mile.

It is remarkable that such accuracy was obtained in view of the fact that most of the “known” facts and his observations were incorrect: (1) although it is true that the sun at noon is directly overhead at the Tropic of Cancer on the day of the summer solstice, it was erroneously concluded that Syene lay on the line. Actually, Syene is 37 miles to the north; (2) the true distance between Alexandria and Syene is 453 miles and not 500; (3) Syene lies 3° 30′ east of the meridian of Alexandria; (4) the difference of latitude between Alexandria and Syene is 7° 5′ rather than 7° 12′ as Eratosthenes had concluded.

Another ancient measurement of the size of the earth was made by the Greek, Posidonius. He noted that a certain star was hidden from view in most parts of Greece but that it just grazed the horizon at Rhodes. Posidonius measured the elevation of the same star at Alexandria and determined that the angle was 1/48th of circle. Assuming the distance from Alexandria to Rhodes to be 500 miles, he computed the circumference of the earth as 24,000 miles. While both his measurements were approximations when combined, one error compensated for another and he achieved a fairly accurate result.
Especially with regards to the case of maps, the question is: How could people that did not have access to satellites and/or other similar means, to pinpoint locations with such accuracy and create these “geodetic” or however else they are called, relations?

Jain Chalkboard, hand-drawn February 2025 for my social media audience of 1.6m followers.
Also, credit is given to Carl Sagan who first documented this topic on Eratosthenes in his TV series called Cosmos. In my video, I am holding a map of Egypt that is being held and explained by the late Carl Sagan.