Verticals if you can not get horizontal antennas "high"

I have recommended that horizontal antennas be a half wave above ground for best performance. For the lower bands, this states to be a problem for most of us. While a half wave on 20 meters is only 33 feet, on 40 this becomes 66 feet and on 80 meters this is now 120 or 130 feet! Most of us can only have "low" dipoles on 80 or 160 meters. In order to have a good performing antenna on these bands, especially for DX, we have to either put up a very high tower or use some form of vertical. It is amazing how much improvement a simple bent vertical or inverted L antenna can be over a dipole at the same "low" heigth.
The low dipole does a great job for distances up to maybe 500 miles on 160 or 80 meters. If you want a reasonably strong signal at greater distances, some form of vertical is porbably your answer.
I was able to hang a half square antenna for 80 meters between two pine trees. I was not able to get the vertical elements the desired 66 feet high so I bent the bottom ends. This does not affect the performance much at all because, in this antenna, the two ends at the ground are open. (Not connected to anything). What do we know about the open ends of an antenna? The current is zero at the ends. So the last 5 or 10 feet on a 80 meter antanna wire dies not cary much current. If there is not much current we find that there is not much radiation. Most of the radiation from an antenna is from the high or higher current areas on the antenna. The maximum height of my half square antenna is only about 50 feet. For working DX, say Europe for examply, I would need an 80 meter dipole about 130 feet high to equal the 50 foot high half square! The antenna mocking programs such as EZNEC predict this, and I can say that my experience with this antenna has verified that prediction time and again.