Measuring the Earth

On Monday my class used Eratosthenes method, and some simulated observations in Stellarium, to measure the diameter of Earth.  The result is amazingly accurate considering the technology available to Eratosthenes (which consisted of, basically, a stick and somebody who was willing to walk long distances and count his steps).

Here's the idea: Eratosthenes had heard that in the city of Syene (now Aswan, Egypt) on the summer solstice, sunlight would reach to the bottom of a very deep well.  This implies that the Sun gets directly overhead at Syene on the summer solstice (and therefore that Syene is on the Tropic of Cancer).  So a gnomon (a vertical stick) would cast no shadow at all in Syene on that day.  However, Eratosthenes lived in Alexandria, Egypt.  He could measure for himself the shadow cast by a gnomon in Alexandria on the summer solstice.  His measurement, plus some trigonometry (really the method of chords, because the Greeks didn't have what we now call trigonometry) told him that the sunlight at Alexandria was coming in at about 7 degrees off of vertical.

Eratosthenes assumed that the sunrays hitting Alexandria and those hitting Syene were parallel (and we investigate why this is a reasonable assumption, although there are some problems with the finite angular size of the Sun).  In that case, the 7 degree difference in the direction of the Sun rays relative to vertical must come from a difference in the latitude of the two locations: Alexandria must be 7 degrees North of Syene.  Eratosthenes believed that Alexandria was DUE NORTH of Syene, and he knew that the distance between the two cities was about 500 miles (in modern units) because someone had paced it.  So he decided that 7 degrees around Earth corresponds to a distance of 500 miles.  Proportionality implies that 360 degrees corresponds to a distance of just under 26,000 miles, which would be the circumference of Earth.  Divide by pi to get the diameter, a bit more than 8000 miles.

The activity guides the students through this procedure, and lets them make measurements along the way using Stellarium.  They can set their location to Alexandria or Aswan, and set the time to 200 BC, and measure the altitude of the Sun at local noon on the summer solstice in both locations.  But in the activity we also investigate Eratosthenes assumptions.  He assumed Alexandria and Aswan had the same longitude, but we find that this is not true because local noon occurs at a later time in Alexandria than in Aswan, which implies that Alexandria is a bit to the west of Aswan.  He also thought that the Sun's altitude in Syene on the solstice would be 90 degrees, thus giving no shadow.  In fact, it is 89.5 degrees which would give a 0.5 degree shadow (hard to detect, to be sure, but 0.5 degrees is not 0).

The really remarkable thing is that these two errors nearly cancel each other out.  I don't have my students work through this, but I probably should.  It's not that hard to see how it works.  Local noon in Alexandria is roughly 10 minutes later than in Aswan, which implies a 2.5 degree difference in longitude.  The real difference in latitude is 6.5 degrees, not 7 degrees.  But if we now combine the 6.5 degrees north with the 2.5 degrees west using a simple Pythagorean theorem approach (which isn't really correct, but it is a good approximation because the angles are small and we are close to the equator) we find that the angular distance over Earth's surface from Syene to Alexandria is the square root of (6.5 squared plus 2.5 squared), which works out to 6.96 degrees, or essentially the same as the 7 degrees that Eratosthenes thought it was.

Sometimes it is better to be lucky than good, although I would argue that Eratosthenes was both.  It's such an elegant solution to the problem of measuring Earth's diameter.  I think my students enjoyed this one.  It was a bit short, but maybe next time I'll have them think their way through the impact of the two errors mentioned above so that they can see why they basically cancel each other out.  It will be challenging for them, but if I can get them to do it I will be very happy.  That kind of analysis of error is a pretty high-level scientific thinking skill.