If the radius of the Earth is r, and you have height h, then from the top of your head to the center of the Earth to the horizon and back to the top of your head is a right angle triangle. The hypotenuse is r+h, one side is r, and the other must be sqrt(2rh + h*h). Under the assumption that r is much bigger than h, that means that the distance to the horizon varies as the square root of the height.
This assertion is surprisingly easy to check with ships. Furthermore if you pay close attention on a commercial aircraft, you can actually see that the horizon is a little bit below level!