How to calculate the horizon distance?

What does it mean to calculate the horizon distance? This means finding out how far from the observer the horizon line is? In other words: how big is the radius of that circle, in the center of which we see ourselves on level ground? Let's answer this question. When we watch a ship emerging from under the horizon from the seashore or a large lake, it seems to us that we see a ship not at the point where it really is, but much closer, at the point where our line of sight slides along the convexity of the sea . When observed with the naked eye, it is difficult to get rid of the impression that the ship is at this point, and not further beyond the horizon.

However, in the telescope this difference in the distance of the vessel is perceived much more clearly. The pipe doesn’t show us objects that are close and distant equally clearly: in a pipe that has been set afar, close objects can be seen vaguely, and back, a pipe that has been set on close objects shows us in the distance in the fog. If, therefore, to direct the pipe (with a sufficient magnification) to the water horizon and set it so that the water surface is clearly visible, the ship will present itself in a blurry outline, thus revealing its greater distance from the observer. On the contrary, having installed the pipe so that the outlines of the ship half-hidden under the horizon could be clearly seen, we notice that the water surface at the horizon has lost its former clarity and is drawn as if in a fog. English astronomer Proctor, who noticed this instructive phenomenon, said on this occasion: "Everyone who happened to make such an observation unanimously claimed that, no matter how strong their confidence in the earth's sphericity, they found in this observation the most convincing confirmation of this truth".

Let us now try to calculate the horizon distance, knowing the magnitude of the observer’s elevation above the earth’s surface.

The task is reduced to calculating the length of the segment CN (Fig. 1) of the tangent drawn from the observer's eye to the earth's surface. The square of the tangent is equal to the product of the outer segment (h) of the secant over the entire length of this secant, i.e. on h + 2R, where R is the radius of the globe. Since the elevation of the observer’s eye over the earth is usually extremely small compared to the diameter (2R) of the globe, making up, for example, for the highest elevation of an airplane, only 0,001 of its share, 2R + h can be taken equal to 2R, and then the formula will be simpler:

CN2 = 2Rh

Hence, the horizon range can be calculated using a very simple formula:

horizon distance = (2Rh)1/2,

where R is the radius of the globe (about 6400 km), and h is the elevation of the observer's eye over the earth's surface. And since 64001/2 = 80, the formula can be given the form:

horizon distance = 80(2h)1/2 = 113(h)1/2,

where h must necessarily be expressed in parts of a kilometer. For example, let's calculate how far a person can stand on a plain?

Considering that an adult's eye rises above the ground by 1,5 meters, or by 0,0015 kilometers, we have the horizon range = 113(0,0015)1/2 = 4,38 km.