### Estimating the radius of the Earth - The medieval way

I had an interesting discussion with a couple of friends yesterday and we stumbled upon the topic of various astronomical scales, ranging from the radius of the earth to the distance to the sun and how ancient astronomers estimated them. Specifically, let's talk about the radius of the Earth and how it's measured.

But before we get to it, I have a few tangential questions to ask you. What causes seasonal changes on Earth and what is special about the Tropic of Cancer and Tropic of Capricorn i.e 23 and 1/2 degrees north and 22 and 1/2 degrees south? Quite a few people think that the apparent change in the distance between the Sun and the Earth as the Earth moves along it's elliptical orbit is what causes the seasons. In reality, if you look at the numbers, you will see that the ratio of the light that earth receives at it's perigee is 0.94 times that it receives at it's apogee, not large enough to change the temperature significantly. What causes the seasons is in fact the tilt of the Earth's rotation axis with respect to the plane of revolution of Earth around the sun. Because the Earth's rotation axis is tilted with respect to the normal to the plane of revolution, you can see that at one point of the revolution, the southern hemisphere sees the sun above head where as the northern hemisphere sees it near the horizon and at the opposite point of the revolution, it is vice-versa.

Now, as you can see, on the summer and winter solstices, the 23 and a 1/2 degree north and 23 and a 1/2 degree south latitudes see the sun right above head! You can say that on these days, a person at the Tropic of Cancer or the Tropic of Capricorn will not cast any shadow! That is the reason why these two latitudes are special geographically!

Coming back to the question at hand, how do we use this information to estimate the radius of the Earth? Well, it so happened that in the medieval ages, someone found that a well in Syene, south of Alexandria, cast no shadows on the walls of the well or on the water in the well on one specific day of the year. Eratosthenes, the then librarian of the Alexandrian library heard about this and devised a method to estimate the radius of the Earth.

If we place a vertical pole on the ground, we know that it casts a shadow, the length of which depends the angle of the sun's rays i.e the shadow will be the longest when the Sun is near the horizon and the shortest when the sun is above head! If we place the same pole in Syene on that momentous day, we would see that the pole doesn't cast a shadow. We can understand from this that Syene was on the Tropic of Cancer and that the momentous day was the summer solstice! We can now move to an upper latitude, say to Alexandria, and measure the distance between the latitudes along the longitude. And if we perform the same experiment, of placing the pole vertically on the ground, we will see that a shadow is cast by the pole, the length of which we can measure. Now, using trivial geometry, we can estimate the angular separation between Alexandria and Syene. And because we already measured the distance between them, we can estimate the radius of the Earth! Take a look at the following pictures and you'll hopefully see it better!

Using this method, Eratosthenes estimated the radius of the Earth to an accuracy of 2-15%, which is not at all bad! So there we are, in the first step towards quantifying the distances in the solar neighborhood!

I used the wiki page on Earth to estimate the ratio of apogee to the perigee and the wiki page on Earth radius to know the estimates Eratosthenes came up with (and to know how to spell Eratosthenes!). I borrowed the images from here and here.