I think the original questioner is still scratching his head. If so, try this.
The earth isn't really spherical, but we're going to pretend it is. So no use claiming to be super accurate. I guess a sphere with 3440 nm radius is about as good as any. We're also ignoring refraction.
Also we're ignoring the earth's spin-- so a vertical line points to the center of the sphere. And a horizontal line is defined as a line perpendicular to a vertical line. So a horizontal line doesn't point to the horizon-- right?
If a horizontal line just touches the earth's surface, and you are on that line X nautical miles from the point where it touches, how far above the earth's surface are you? If you want the answer in feet, it's 0.883 times X squared. That's correct to within 0.2% beyond 200 nm.