OK, I have the proof. I'll be using an image I drew myself for reference, as Tom's Image didn't highlight the correct areas to solve the problem.

The goal here is to find the length

*a*, which is the vertical difference between

*A* and

*B*.

Let's use the above model, where

*O* is the centre of the Earth,

*A* is our standing position, and

*B* is our viewing position. The diagram shows us that (assuming that the earth is a perfect sphere for this example, otherwise it becomes far too complicated,) the radius

*r* is the same along the line

*OA* as

*OB*. Since we know the length of the arc

*AB* (which I just realised I forgot to label, let's just call it

*l*) we can find the angle

*θ*, (in degrees, it would be

^{l}/

_{2πr} x 360.)

Now, let's assume the position of the Earth is at (0,0). (Note that for this example we do not need to worry about moving in the

*z* direction, because the example assumes no movement in this direction, and thus can be ignored.) Now using simple geometry, We can define the point A (0,

*r*) and B (

*r*sin

*θ*,

*r*cos

*θ*).

Now we know that the length

*a* is the difference in height between

*A* and

*B*. So by defining the coordinates, we get,

*a* =

*r* -

*r*cos

*θ*.

And this is the difference in height between any 2 points on an arc of known length.

(Please note that I have not yet tried the numbers given in the example so I don't yet know if this will help my argument or not. I did this deliberately so my proof would not be biased in any way, thus the use of unassigned letters to show the answer.)

I will apply the numbers from your 30 mile example and post it in a new post.