This is a more rigorous demonstration of what 8" per mile squared means.

θ = 1/3961 = the earth's curvature.

The arc subtended by that angle (with respect to the earth's center) is Rθ = 3961×3961 = 1 mile.

The line segment that connects your position and the horizon (passing through the earth) is c = 2R² - 2R²cosθ.

If your horizon is 1 mile away, then your observation altitude h is given by the equation: h = √(d² - c²).

In terms of c, the preceding equation can be rewritten as c = √(d² - h²)

Your line of sight distance to the horizon is given by the equation: d = √[(R + h)² - R²] which reduces to √(2Rh + h²)

We call replace d in the equation c = √(d² - h²) with √(2Rh + h²) to give us c = √(√(2Rh + h²)² - h²) = √(2Rh)

Therefore, √(2Rh) = 2R² - 2R²cosθ

and solving for h,

h = (2R² - 2R²cosθ)²/2R

This equation relates your elevation with the earth's radius and the angle θ subtended by the ground go the horizon from the earth's center.

When the distance to the horizon is 1 mile, θ = 1/3961, simply the unit arc.

h = (2×3961² - 2×3961²cos(1/3961))²/(2×3961) = 0.000126230748261746 ≈ 7.99"

If we double the arc length to 2 miles, then 2(1) = 2R,

h = (2R² - 2R²cos2θ)²/2R