Here... draw a Cartesian coordinate on paper. Put eight dots all around the origin with a ninth dot being the origin. (should be {0,0}{0,1}{1,1}{1,0}{1,-1}... etc.) now put 16 dots around that ({0,2}{1,2}{2,2}{2,1}... etc.) If we are at point {1,1} and each and every dot moved away from each other at the same amount, the visible distance between {-2,1} and {-2,-1} would be further apart from our point of view. The distance between {2,1} and {2,2} would be smaller than that amount.
First of all, if you're an observer at point {1,1}, measuring the angular separation between objects on the sky(i.e. how far apart are e.g.{-2,1} and {-2,-1}), then it remains constant when you increase all the distances between points by an equal factor(for example, multiplying all coordinates by 2 does that). Also, the angular separation would be equal to that between points {2,1} and {2,2}, which is in both cases 45 arc degrees(π/4 radians).
As you can see, there's no point in measuring the tangential velocity of galaxies, if you're trying to find out about the expansion of the universe. All that you'll find is relatively small movements due to the gravitational attractions of each body's close neighbourhood.
We would also notice a difference in redshift from our point {1,1} and {2,2} than we would from {-2,-2} because... if all points were moving apart from each other, the distance would be three times as much between those points and the shift should be much greater. If you correlate a set of points all around us, you should be able to determine if we are on an edge, near the origin and/or where the origin is.
See, I've got this strange feeling that one of us misses the other's point completely, as we both use the same argument to both attack and defend the theory in question.
So, yes, that's true, it's exactly what we see - the farther away is a galaxy from ourselves, the higher is it's redshift. Just as it's expected from the universe that is expanding in all directions. Now, if you'd look at all those redshifts, you'd inevitably notice that all points are moving away from you, which would suggest that you're at the center of the universe, according to your line of thought(as far as I understand). Choosing any one of the infinitely numerous points in the universe as your vantage point, and using the same line of reasoning as before, you must come to the conclusion that there are infinitely many centers of the universe.
Does this make any more sense now?
ninja pseudoedit:
But nobody can take that data and determine, based on differences in movement, what direction the majority are heading and tangent from that a guess to where they all came from?
Ok, I understand your concern. Actually they don't move by themselves, it's space that is getting bigger.
Yeah, and now back to the circular thought that I mentioned before on how some people say it's not possible to measure because the space between expanding causes light passing through it to shift and this is where the original argument came from. (convenienthandwave)It's impossible to measure!(/convenienthandwave)
But what are you talking about? What do you want to measure? The distance between galaxies? It is measurable, indirectly of course, by comparing their velocities(redshift) with the Hubble's constant(D=V/H).