Wednesday, May 7, 2014

Eratosthenes' Estimate of the Earth's Circumference

Eratosthenes as head of the Museum in Alexandria, Egypt had access to the best geographical information of his time and made an estimate of the Earth's circumference. Although only fragments of his writings now exist we know through later writers such as Cleomedes that from the nearly vertical shadow in a well in ancient Syene (modern Aswan) and the shadow of an obelisk in Alexandria Erathosthenes concluded that the difference in latitude was 1/50th of the circumference of the Earth. Assuming that the distance between Alexandria and Syene was roughly 5000 stades he arrived at an estimate of 250,000 stades for the Earth's circumference. We can check what he did.

For Google Earth positions we can use a spice market in Aswan as the location of the well in ancient Syene and the location of Cleopatra's Needles in front of the Caesarium in ancient Alexandria as the location of the obelisk. From this data we get 1/50.6 for the fraction of the Earth's circumference and 4675 stades as the great circle distance between the two positions. Our estimate of the Earth's circumference is found to be off by 10% from the theoretical value based on 10 stades per arc minute. It is not unreasonable to accept Eratosthenes' estimate in the way that it was presented.

What happens if we apply a little more mathematics to the problem by including the deviation of Syene from Alexandria's great circle meridian? Instead of 7.59 degrees separation between Alexandria and Syene we have to use 7.08 degrees for the separation in latitude. Our conversion factor from angle to degrees tells us that this separation corresponds to 4372 stades. So our error from the theoretical model is reduced to about 3% and this is approximately the error in the Egyptian foot.

We have to consider the possibility that the Egyptian foot was intended to be a geometrical unit of length but that it's value was slightly off.