Math CentralQuandaries & Queries


Question from Kevin, a parent:

Working on a project for a science fair.
To prove the earth is round without the ability to get way above the surface.
How could we set up and experiment to see the curve.
My thought is using a telescope at the beach?
Form what we see so far the curve is 8 inches per mile?
So thought is set up a telescope and a target 1 mile away?

Hi Kevin,

Curvature has dimensions of inverse distance (it goes with radius, but the bigger the radius the smaller the curvature.) Thus, it's not inches per mile but inches per mile squared. $\left(\large \frac{1}{D} \normalsize = \large \frac{D}{D^2} \right)$ This means that longer distance makes it much easier.

Most easy ways to measure the earth's curvature directly use a water surface, as a surface of equal gravitational potential. The sea or a big lake work well. Watching a ship go "hull-down" at the horizon gives evidence of curvature. Measuring how far the horizon is as a function of moderate height gives a more interesting measurement. At an eye height h (in meters), the horizon (over water) is about $3.57 \sqrt h$ kilometers away. We can actually calculate the radius of the earth from the value of the constant.

This ignores refraction, however. Light bends from less dense media into denser media. This is most easily seen in the silvery "mirage puddles" when hot (hence less-dense) air near a road surface bends light away from the road. But, without such intense local heating, the greater pressure of air near the earth's surface can bend light the other direction, making it follow the earth's curvature. As a result of this, the sun appears to slow at the horizon and set a few minutes later than astronomy would predict.

If the air near the earth's surface is colder than that aloft, the effect can be much stronger and light may bend as much as the earth's surface does. In such a situation, the earth appears flat or even slightly bowl-shaped, retreating objects never appear to reach the horizon (they just fade and become tiny), and land normally beyond the horizon may appear as strange flickering walls and towers (the "Fata Morgana" mirage.)

Good Hunting!


The approximate formula for the distance of the horizon from the viewer is easier to remember when given in feet and miles:

If the viewer’s eyes are f FEET above the ground, the horizon (over water) appears to be $\sqrt {\left( \frac{3}{2} f\right)}$ MILES away.

Thus when the eyes are 6 feet above the ground, the horizon is about 3 miles away.


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