Measuring the size of the Earth

Another Greek geek!

The ancient Greek scientists were extremely clever at figuring out how the world works. Eratosthenes lived in the third century before Christ and did an ingenious experiment to measure the Earth's circumference.

Eratosthenes' measurement of the circumference of the Earth

At noon on the summer solstice, the sun is directly overhead in Aswan (then called Syrene). Alexandria is about 500 miles due north of Aswan. Eratosthenes measured out this distance and then, at the moment of the solstice, measured the shadow of a vertical stick in Alexandria. From the ratio of the shadow's length to the height of the stick, he deduced the angle of the sun's rays to be 7 degrees off vertical. We can see from the above picture that this is also the difference in latitude between the two cities. 7 degrees is about 1/50th of a full circle, so he just multiplied the 500-mile distance by 50 to get the circumference of the Earth. In this rough calculation, we get 25,000 miles, which is close to the modern accepted value.

Do you see the flaw?

This is so simple and elegant! But what might bother you about it? Eratosthenes assumed that the sun is extremely far away compared to the Earth's size so that all the sun rays are parallel. That's why the sunray angle equals the latitude difference in the above picture. Suppose that the sun were much closer and the Earth were flat, as shown below. We can reproduce the 7-degree sunray angle if we assume that the sun is about 4000 miles away.

Alternative explanation of Eratosthenes' result: A flat Earth with the sun 4000 miles away

How do we decide which picture is right?

There are also many cases with the sun distance and Earth's radius having in-between values. Need we give up and say that this kind of experiment can't decide which picture is right? Fortunately, the answer is no. Let's see what is expected to happen when we measure the noonday shadow at various places at different latitudes.

Plots of the sunray angle versus latitude difference for the distant sun over a spherical Earth and for the local sun over a flat Earth

The above plot shows how the sunray angle predicted by each model changes as we go farther and farther north. For small latitude differences, they give close to the same sunray angle. As the latitude difference gets larger, they diverge. The flat Earth model with a sun 4000 miles away gives a significantly smaller sunray angle at high latitude differences.

What does the data tell us?

A few years ago the Noon Day Project at Stevens Institute of Technology had school children in various places around the world reproduce the experiment. I also did my own version. Let's add this data to the plot.

Measured data points added to the sunray angle plot closely match the distant sun model.

The blue dots are measurements by students at the equinox, when the equator is the point where the sun is overhead. Eratosthenes' summer solstice experiment is the green star. In order to push to as large a latitude difference as possible, I did the experiment at the winter solstice, when the sun is overhead at the Tropic of Capricorn. At my location in Boston, this gives a latitude difference close to 65 degrees; this is the circled data point up in the right corner. Except for a couple of student outliers, the data points hug the straight line, in agreement with Eratosthenes' assumption of parallel sun rays. The flat Earth model with a close sun gets it wrong; it predicts that the data points should follow the drooping curve.

A note to science teachers

Your students can do a multi-latitude experiment without needing to travel to faraway places. Just do the experiment at three different times of the year. At the June solstice, the latitude difference will be the difference between your latitude and the Tropic of Cancer (23.5 degrees N). At the September or March equinox, the latitude difference will be your latitude. At the December solstice, the latitude difference will be the difference between your latitude and the Tropic of Capricorn (23.5 degrees S).