## Sunday, January 03, 2010

### A leap into the unknown

Let me rehash here a story I told in Walking Zero, of how Eratosthenes measured the size of the Earth. I want to draw on your collective experience.

Eratosthenes lived in the the city of Alexandria at the mouths of the Nile in Egypt in the 3rd century B.C.E.. He was the librarian of the great book (scroll) repository at Alexandria, the greatest library of its time. We know almost nothing about him personally, other than his skills and a geographer and astronomer. The story of his measurement of the Earth is often told.

Here is the relevant diagram. Eratosthenes had heard from travelers of a deep well at Syene, some hundreds of miles down the valley of the Nile, where at noon on midsummer day you could see the Sun reflected in the bottom of the well -- which implies, of course, that the Sun is directly overhead. He knew from his own observations that the Sun is not directly overhead in Alexandria at the same time. And, as the diagram shows, the reason is clear if the Earth is a sphere.

At noon on midsummer day Eratosthenes measured the length of the shadow cast by a pillar at Alexandria whose height he had previously determined, and therefore the angle made by the Sun's rays with the vertical at Alexandria. As you can see from the diagram, this is the same as the angle subtended at the center of the Earth by Alexandria and Syene. The angle he measured was one-fiftieth of a circle. Therefore the distance from Alexandria to Syene is one fiftieth of the circumference of the Earth. From the experience of travelers, he knew approximately the distance between the two cities, and therefore the Earth's circumference. His result, we now know, was remarkably accurate.

Again, this story is told in every astronomy book. But I have yet to come across a telling of the story that notes one crucial point. The diagram above assumes the Sun's rays are parallel; i.e. that the Sun is very far away compared to the size of the Earth -- a remarkable assumption. As you can see from the following diagram,Eratosthenes might just as well have assumed the Earth was flat and used his angle measurement to determine the distance to the Sun.

Presumably he had good reasons to believe the Earth was a sphere {the changing elevation of stars as one travels north/south, the masts of ships on the horizon, the shadow of the Earth on the Moon during eclipses, etc.). But even then, was it fair to assume the Sun so far away? This was, on the face of it, a bold leap into the dark, later confirmed by another Alexandrian, Aristarchus, in what I believe to be the most ingenious work of science in antiquity (see Walking Zero).

At the heart of Eratosthenes' brilliant deduction of the Earth's circumference is an outrageously bold assumption about the size of the cosmos -- one that I have yet to see mentioned in accounts of his story. Does anyone reading here know of a published account that gives those parallel rays due discussion?

Using the well and shadow to figure out the size of the Earth was ingenious. Imagining the Earth in a universe of staggering dimension was perhaps an even greater contribution to the progress of human knowledge.