My geometric argument that beyond distances of several hundred,
perhaps 500 miles, the vertical puts down a stronger signal and
receives stronger signals than the horizontal dipole cannot be
disputed.

If you can't be heard at 1000 miles or more using a dipole, you are
more likely to be heard using a vertical regardless of what antenna
the other fellow is using to receive. At great distances you are much
more likely to be heard using a vertical at the same average height
above its surroundings.

Signal to noise ratio does matter of course.

Local noise level is much greater than received from distance sources
for obvious reasons. Local noise is vertically polarised. It comes in
via groundwave.

Noise from a distance is randomly polarised. It comes in via the
ionosphere. So in towns and cities, with buildings wiring, overhead
power and phone lines, where most of us live, the vertical collects
more local noise. In the wide open countryside both types of antenna
tend to perform equally well on randomly polarised, distant noise
levels.

With distant noise and interference and distant signals, both types of
antenna result in the same signal to noise ratio in the receiver. But
the vertical antenna receives the stronger signal plus noise. If the
internal receiver noise is greater than the received signal plus noise
then the vertical antenna will win the contest.

However, there is another effect which sometimes gives the dipole the
advantage. It is multi-hop propagation.

The angle of elevation of the radio path increases with the number of
hops involved. The number of hops depends on the sun-angle and day or
nighttime. Across the States or across the Pacific, for example, the
propagation loss can be much less with 2 or 3 hops than it is with one
or two hops. Waves sometimes bounce between the F2 and E layers. The
increase in elevation angle favours the horizontal dipole. And how
many amateurs know the number of hops involved at any point in time?

But what eventually favours the vertical over the dipole is their
respective service areas. The service area covered by the vertical is
many times, far greater than the dipole and so is the world wide
distribution of radio amateurs and short-wave listeners.

We have now returned to the simplistic but precise Geometry of the
ancient Egyptians and Greeks. ;o)
----
Reg, G4FGQ.