200 Proof: Beyond the Horizon

Instinctively, we all know what the horizon is. Not everyone understands why a horizon exits though.

6) If Earth were a ball 25,000 miles in circumference as NASA and modern astronomy claim, spherical trigonometry dictates the surface of all standing water must curve downward an easily measurable 8 inches per mile multiplied by the square of the distance. This means along a 6 mile channel of standing water, the Earth would dip 6 feet on either end from the central peak. Every time such experiments have been conducted, however, standing water has proven to be perfectly level.

Standing water does not have a central peak on a ball Earth. From wherever an observer is standing, the world curves downward at a very gradual slope. The equation that the author is using is:

Curve downward = 8 inches/mile * distance² [where distance is in miles]

8 inches/mile * (3 miles)² = 72 inches = 6 feet [3 miles because he is using the center]

Unfortunately this leads to a problem when you graph out the equation since it is an exponential loss of height:


If this was the case the Earth would continue away for every. Since the Earth is a ball, it means the rate of curve away has to change and eventually become negative (curve back toward itself. Intuitively, when the distance is small, the loss of height from the curve is almost zero (like the Earth is flat), but increases toward the 1/4 lap mark. At the halfway mark, the curvature needs to reverse itself so a loop can happen:


Really, the problem is the formula needs to be more accurate. The one cited works during the first portion, and the 6ft of dip over 3 miles is very close. The way to calculate it requires a trigonometry refresher. First, the Earth has a diameter of 7917.5 miles (Radius = 3958.75 mi) and a circumference of 24,900 miles. A sphere has 360 degrees along one of the great circles (biggest possible circle).  One mile of distance on the surface therefore is:

1 mile ÷ 24,900 mile * 360 degrees = 0.01446 degrees

Next is where the trigonometry comes into play. Remember
[ sin(a) = opposite length / hypotenuse ] when dealing with right triangles.

Proof6-calculate height lost

R is the constant Radius (3958.75 mi). The number of miles walked on the surface will give you “a” (0.01446 degrees / mile). The base can be calculated from the isosceles triangle as

base = 2 * R * sin(a/2)

There are 180 degrees in a triangle, so (180-a)/2 gives the other two angles in the isosceles triangle. The angle “b” therefore is found to equal 90-(180-a)/2 or equal to a/2.

Finally, the h can be solved as h = base * sin(a/2) = 2 * R * sin(a/2) * sin(a/2). That is really just because math is fun. At about 1000 miles distance it will the author’s equation will be wrong by a little over 0.63 miles.

The real question is: does standing water remain level or is there a slope? and have all experiments demonstrated that it is flat?

The “Bedford Experiments,” which the author is referencing when performed with the flag at 5 ft does show that the flag is visible further than would be expected. This is attributed to refraction. In fact, when the test was performed with the flag at 13 feet, it demonstrated that it was shorter and in line with the curved Earth. It is inaccurate therefore to say all tests have demonstrated a flat earth. It is also something that is commonly considered for surveryors.


You are probably familiar with how refraction creates an optical illusion where your drinking straw looks broken at the air/water junction. Refraction is what is at play when a fish looks like it is in a slightly different part of the fish tank. Refraction causes the mirror like mirage on the hot road in the distance. It also can make an object appear higher than it is as the light passes through the air.


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