That is actually a myth. The moon does rock back and fourth as it passes overhead, although it does not get to its side.
I recall from an astronomy text that astronomers were able to map out large parts of the far side of the moon long before we had ever sent craft there, but I will have to find the source on where I read that.
I'd like to see that quote - but it's not true. The moon does undergo something called "nutation" in physics and "libration" in astronomy - which is a slight wobble in it's orientation . This allows us to see a small amount past a hemisphere - but not by much.
My favorite image of this is useful to visualise it (this is a set of photos of the moon, taken on consecutive nights over a month and put together into a video):
It's clear that you can certainly see a little way past one hemisphere - but not by much.
As any rate - this isn't Douglas's point here.
He's saying that if the full moon is overhead someplace (lets' say West Africa - at the Equator) - then the people on the equatorial east coast of South America - ought to have a view of the moon that's from a direction that's at least 45 degrees away - and it should be a crescent moon. But in reality, it isn't. They see more or less the exact same thing that is simultaneously being seen in Africa.
Imagine a triangle with the moon at the apex, West Africa and the East coast of South America being the points at the base of the triangle. Since the height of the triangle (in FET) is somewhere around 3000 miles - and the base is around 4,000 miles - the angle subtended between the two lines of sight should be 53 degrees.
That would mean that if the viewer in Africa sees the usual "full moon" view - then the person in South America should be seeing 53 degrees of the "Dark side" and someone out in the middle of the Indian ocean should be able to see at least another 50 degrees of the moon.
But this doesn't happen. The appearance of the moon is more or less the same from all points at the same latitude - and that can only be the case if the moon is MUCH more than 3000 miles away (240,000 miles, for example).