8 Inches of Drop Per Mile Squared
We should see the earth curve at a rate of about 8 inches per mile squared. So looking across a body of water, a distance of say 4 miles, the bottom 128 inches (8 X 4²) of an object should be out of sight. That’s over ten and a half feet.
How that formula was derived
The Pythagorean Theorem says that the sum of the square of adjacent and opposite sides equals the square of the hypotenuse in a right triangle. a² + b² = c².
So we are trying to find the distance the Earth drops down per mile. So the equation would be radius² + distance² = (radius + drop)².
If we are trying to find the drop we can change the equation to √(r² + d²) – r = drop.
If the radius is 3,963 miles and the distance is 1 mile we can solve the equation. √(3963² + 1²) – 3963 = drop. Putting that into a calculator you get drop = .000126 mi. There are 5280 feet in a mile and 12 inches in a foot. So .000126 * 5280 * 12 = 7.98336 in.