PRECISION

Several readers wrote in to point out that the distance to the horizon [AtE Aug 84] is not just the square root of 2Rh (R earth radius, h elevation of vantage point), but of 2Rh + h^2. As a strict exercise in geometry, this would be correct, but when I said the square root should be expanded, I had a good reason.

For one thing, this will reveal the additional term as unimportant: it affects the result by only 1.18%, making the correction quite unnecessary for proving the story of radio contact physically impossible. But more important, going for the next decimal place by geometry gets us into an area where another effect, atmospheric refraction, becomes more important.

It is also pressure, temperature and humidity of the air that make the difference in an exact calculation of the distance to the horizon, for they slightly affect the velocity of light and radio waves, and therefore curve the ray away from the straight line that it would follow in a vacuum.

Why is curvature connected with velocity? Because of a principle discovered by a great French mathematician, Pierre de Fermat (1601-65): light always propagates along a path that will take it from one point to another in the shortest possible time.

To see the significance of this, let a lifeguard at point A spot a swimmer drowning at point B; how can he get to him most quickly? Not in a straight line, because he will spend too much time swimming (from C1) instead of running; nor running to the point C2 from which he needs to do the least swimming, because he will spend too much time running when he could already have been swimming; but somewhere in between (point C): the minimum time, as one can show, is achieved when the ratio of velocities equals the ratio given by Snell's Law of refraction. A ray of light is refracted as if it tried to save a swimmer.

Now imagine the lifeguard in a circular swamp where the going gets continuously tougher toward the center, just as fight propagates a little slower with increasing atmospheric pressure. How will he have to move now when he sees a swamped swamper in the swamp? In a curve, says Pierre de Fermat, bending away from its tangent toward the center -- in fact, Fermat's principle will give the exact formula how to get there.