J. Scott Steflik wrote: Some of the originals I've handled had MORE flex (when I had the balls to test them) than my first training rapier. Looking at detailed measurements of other period pieces and quality reproductions reveals much the same.
The key word here is
some. Certainly there was variation in the manufacture and design of rapiers, just like every other type of sword, and they're not all absolutely the same. I've handled a couple of period originals myself which clearly had no flex at all. John Clements has handled literally hundreds of them and his observation is that the clear majority were very stiff. My personal experience with flexible trainers is as I said: they bounce around in the bind too much and the wobbly point doesn't go where I want it to. Wooden trainers don't have these problems and the techniques feel much more accurate when performed with them in my opinion.
It's also worth noting that this is dependent on exactly what you define as a "rapier". If you include blades that still retain some cutting ability and are thus more flattened, then yes, those will be somewhat more flexible. If you use the term mainly to refer to swords that are basically 4-ft. ice picks, which may have triangular, hexagonal, octagonal, or even star-shaped cross sections for a significant portion of the length, those are not going to flex. Those are what ARMA refers to when we say "rapier," though we understand the definition will never be universal or perfect.
This article contains some beautiful examples:
http://www.thearma.org/spotlight/heymr.htm
And of course this one covers many arguments in much more detail than I can here:
http://www.thearma.org/Youth/rapieroutline.htm