How the diameter of the earth was calculated in ancient Alexandria.

Last night, as our history book club was discussing The Rise and Fall of Alexandria: Birthplace of the Modern World by by Justin Pollard and Howard Reid we handled a question badly. I will try to do better here.

Eratosthenes more than 2000 years ago, not only recognized the earth as spherical, but made a very accurate estimate of its dimensions. He did so basically armed with a stick. The question was, how did he do so.

How did Alexandrians recognize the spherical shape of the earth?

The Light House of Alexandria was one of the wonders of the ancient world because of its great height. Anyone approaching it from a distance from either land or sea would presumably first see its top just over the horizon. Turning toward the tower and approaching more closely, the tower would rise larger and larger until it dominated the view. It seems likely that some great thinker recognized that this would not be true unless the earth was spherical, at least locally around the building. Of course, with that insight, similar examples of items emerging from the horizon to become quite tall would occur in many places.

Similarly, the great ancient astronomers of Alexandria might have recognized that eclipses of the moon were caused by the earth's shadow falling on the moon, and would have seen that shadow as circular.

How did they recognize rotation around a North South axis?

There was little or no light pollution in the ancient world. Not only the astronomers of ancient Alexandria, but common people would have recognized that objects in the night sky as well as the sun rose in the east and set in the west; they would have recognized that the objects directly overhead traveled a long distance from east to west, while the north star was stationary in the night sky. Thus they could easily infer a rotation around the north south axis.

It would have been hard to recognize that it was the earth itself that was rotating -- that is so counter intuitive. For most people for millenniums it was the vault of the sky that they believed to be rotating, but the rotation itself would have been clear.

The recognized the summer solstice

Like people in many parts of the ancient world, Alexandrian astronomers recognized the summer solstice -- the day when the sun reached the northern extent of its annual path, appearing directly over the Tropic of Cancer, which coincidentally passes through Egypt, at what we now call Aswan.

Eratosthenes' method

Eratosthenes knew of the existence of a deep well in Syene (modern Aswan) where at noon on the day of  the summer solstice, the sunlight pierced directly down the deep vertical walls to fall on the water in the bottom of the well. He recognized that at that instant, the sun appeared directly above that point on the earth's surface. We now recognize that the well must have been directly on the Tropic of Cancer.

He must also have realized that the sun was so far from the earth that sunlight could be essentially assumed to fall in parallel rays all over the surface.

He recognized that Alexandria was essentially due north of Aswan, and the distance between Aswan and Alexandria had been carefully and accurately measured (by trained men pacing it off). He would have realized that since the earth rotated on a north-south axis, noon in Aswan and noon in Alexandria would be simultaneous.

He had a pole placed vertically in Alexandria, and at noon measured its shadow. Any sundial would have told him the exact moment of noon. From the length of the shadow and the height of the pole, he could calculate the angle between the vertical pole and the sunlight.

That angle was the angle subsumed by the arc on the spherical surface of the earth between Aswan and Alexandria. Thus he could scale up the distance between Aswan and Alexandria to calculate the total distance of the 360 degree arc of the circumference of the earth. From that in turn he could calculate the radius and diameter of the earth. He did so very accurately.