The Shadow Knows

People have been studying the Earth and the sky for millennia.  It is quite likely that the ability to predict celestial events and to navigate by the sun and stars, at least to some extent, was well understood before recorded history.  The Polynesian people had found and colonized every major Pacific island by 5,000 BC.  The builders of Stonehenge and similar stone circles as well as both Old and New World pyramids did so with a keen awareness of their position in the cosmos.  Ancient cultures throughout the world could clearly observe, measure and calculate many geophysical phenomena.

Not everyone thought the world was flat, either.  The Greek mathematician Eratosthenes (276 - 195 BC) understood it to be a sphere and is generally considered to be the first person to calculate the size of the Earth with reasonable accuracy.  His method was simple and elegant.  It was common knowledge in his time that the sun would be directly overhead at noon on the summer solstice in the Egyptian city of Syrene (present day Aswan). At that moment, the sun's rays would reach the very bottom of a deep well.  Eratosthenes observed, however, that the sun cast a shadow at noon on that day in Alexandria.  He measured the angle of this shadow as 7.2º and reasoned that the distance from Alexandria to Syrene must therefore be 1/50th of the circumference of the world (50 x 7.2 = 360).  His estimate of that distance was 5,000 stadia.  (A stadion is a Hellenistic unit which for which no standard exists, but the actual distances of many historical references yield an average value of 157.7 meters.)

If we use that value, Eratosthenes' circumference would be 39,425 kilometers (50 x 5000 x 0.1577).  That's pretty good!  The accepted polar circumference today is 40,008 km (24,860 mi).  The indeterminate value of his stadion, the accuracy of his distance estimate and the fact that Syrene is not precisely on the Tropic of Cancer all contribute to imprecision.  Nonetheless, the concept is impressive — measuring the angle of solar incidence at two locations simultaneously.

I have always wanted to try Eratosthenes' technique to see if I can measure the world myself.  It's impossible to be in two places at once, but this could be a great classroom experiment.  There are three basic requirements: a means of measuring the angle of solar incidence, two cooperating observers at different locations and the distance between the latitudes of the observers.  Here's how to do it:

1. The height of a vertical object and the length of its shadow are used to find the angle of solar incidence.  Such an object is called a gnomon.  This one is just a 36" length of 1" dowel clamped to a pair of framing squares.  It is important to have the gnomon be plumb and cast its shadow on a level surface.  A more permanent device would be convenient, but it's not necessary.

The length of this gnomon's shadow is measured with a common yardstick.  The tip of the shadow will be slightly fuzzy, so try to measure in the middle of that penumbra.  Measurements should be recorded every few minutes both before and after high noon.  The shortest shadow is the one that should be used.  You now know the adjacent and opposite sides of the angle of interest (the height of the gnomon and the length of its shadow, respectively).  Divide the Opposite by the Adjacent to get the tangent of the unknown angle.  The inverse of that result (the arc tangent) is the angle of solar incidence.  Use a calculator for this, or a site like

2. Cooperative observations need to be arranged between 2 different schools at different latitudes.  The observations used in the final calculation should occur on the same day.  Taking observations over several different days will help overcome weather problems and produce several calculations for comparison.

3. Knowing the surface distance between the latitudes of the observers is the major challenge (just as it was for Eratosthenes).  If the locations are close to the same longitude, actual road mileage could be used.  Deduct any east-west travel as best you can.  This approach may be less accurate, but it's also the most authentic way to do the experiment.  You will be using actual observations for everything.  If that's not possible, you can resort to a computed value.  Use the latitude values of your observation sites as obtained with ThereAbout on this site or by right-clicking on Google Maps and selecting "What's Here?"  Then this simple calculator will give you the distance you need:

North Latitude
South Latitude

Now find the difference in degrees between the solar incidence angles observed on the same day at each site.  Then divide 360 by that difference.  Multiply the result times the distance obtained in Step 3 above to find the diameter of the Earth.  If you have good observations at both sites on several different days, the average may prove to be the best answer.

Here are some questions to ponder:

  • Should observations be made at exactly the same time or when each has the shortest shadow?
  • The shortest shadow may not occur at exactly 12:00 noon (or 1:00 PM Daylight Savings) — why?
  • What would be a practical minimum distance between observation sites?
  • Would 3 or more observation sites help refine your result?

This experiment will accomplish several things besides estimating the diameter of the Earth.  Learning to use common tools, to apply a trigonometric function, to make careful observations and to collaborate with remote colleagues are all hidden benefits of doing hands-on geography.  No wonder Eratosthenes enjoyed it so much.


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Posted in Activity, Geography, History on Sep 26, 2016