The RE isn't a horizontal slope which slopes downwards at 8 inches for every mile. That's just the drop in the first mile. The RE is an arc of a circle which continues to
bend away from the observer as distance increases.
Suppose that the earth is a sphere with a
radius of 3,963 miles. If you are at a point P on the earth's surface and move tangent to the surface a distance of 1 mile then you can form a right angled triangle as in the diagram.
Looking over a distance of
1 mile, we can use the theorem of Pythagoras:
a
2 = 3,963
2 + 1
2 = 15,705,370
and when we square root that figure we get a = 3,963.000126 miles
Thus your position is 3,963.000126 - 3,963 = 0.000126 miles above the surface of the earth.
0.000126 miles = 12 in * 5,280 ft * 0.000126 mi = 7.98 inches
Hence after one mile the earth drops approximately
8 inches.
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Looking over a distance of
6 miles, we can use the theorem of Pythagoras:
a
2 = 3,963
2 + 6
2 = 15,705,405
and when we square root that figure we get a = 3,963.004542 miles
Thus your position is 3,963.004542 - 3,963 = 0.004542 miles above the surface of the earth.
0.004542 miles = 5,280 ft * 0.004542 mi = 23.98 feet
Hence after six miles the earth drops approximately
24 feet.