I'm thinking about paring a Campy Centaur 10s Medium cage RD with a 12/25 cassette. Is there any issue with that combo? Should I use a Short cage RD?

It will work.

Common "wisdom" will claim that longer cages shift slower, but that is not the case. The shifting occurs at the upper pulley, not the lower one. I've got an old Chorus 10 medium cage on my 11 speed winter bike. with a 50/34 and 12-27.

true, but short cage will always look better and can handle pretty much anything up to 29.

A short cage will handle the wrap of a 13-29 10 speed cassette on some bikes, but not all. A commonly overlooked FACT is that bikes with the perfect chainstay length gain 2T of wrap over those that have the worst case length. The perfect length is one that requires a 52, 53, 54 inch chain length when the the big ring, largest cog and chainstay length are put into an accurate chain length calculator.

Some people get the idea that because they got by using a 13-29 on one bike means that it will work the same on all bikes. Not the case.

Campy says short cage on all but the 13 x 29 with either the standard or compact double, medium cage if you want to run 13 x 29 with either type of double. I'm not saying you couldn't possibly run a short cage with a 13 x 29, just quoting what Campy says. So for the OP, you don't NEED a medium cage with your 12 x 25 cassette even if you're running a compact crank, but other than a slight aesthetic penalty I don't think there's any problem at all with running a medium cage. On my Campy equipped bike, I run a compact crank, 13 x 29 cassette and a medium cage RD. It works great.

There is only an "aesthetic penalty" in certain people's eyes. Those people are not the ones that you need to worry about impressing. The medium cage can do everything that the short cage can, plus more, it is therefore functionally better, and that is all that matters.

+1

I think short cage derailleurs look kind of funny, myself. I think the "short cage looks better" syndrome is a side-effect of seeing pro riders use short cages, and seeing expensively spec'd bikes mounted with short cages when a long cage would work as well and offer more versatility.

Something else I just thought of: There is actually slightly less resistance in using a longer cage derailleur--not so much that would affect performance on any level, but long cages feel smoother--mainly because the chain is twisting less rapidly around the pulleys. With a long cage, the chain doesn't need to weave as sharply, so the forces on the chain are more tangential to the pulley.

I didn't believe it until one of the bike mechs had me hop on a two identical bikes, one with a triple and long cage, the other with a standard double and short cage... the long cage felt just a touch more fluid.

nope. people will say it won't quite shift as well (crisp), but you probably won't notice.

DArthurBrown, I don't think that is correct; the diameter of the pulleys is the same so the amount of bend/unbend that each chainlink experiences going through two pulleys is the same.

I drew a diagram of it, and it appears that DArthurBrown is correct, the chain would have to bend less to get around the pulleys of a medium-cage derailleur than a short-cage. The chain is marked in red, and the line angles are copied for direct comparison on the right.

I think you should draw a diagram from a different perspective. Draw one looking from the rear (or front) of the bike. To consider all the bending the chain undergoes, you'd also have to show the position of the two (or three) front chainrings as well as the freewheel. I think bending of the chain in that plane might be a more important factor.

I agree that bending a chain in the lateral plane is certainly more important than bending in the vertical plane. I believe lateral bending would be extremely similar for short and medium cages, although if anything, the short cage would require a slightly sharper angle of lateral deformation, but the difference would be minimal most of the time.

However, all of these things are only affecting the slack lower part of the chain, not the upper part where the power transfer occurs. This makes it even less likely that these differences would have any measurable effects. This still means that no-one has identified any functional advantage of the short-cage version, and the extra versatility of the medium cage means that the medium cage should still be preferred whenever given a choice.

It isn't the radius of the bend around the pulley that I was referring too, although smaller pulleys would increase resistance as you are describing. The diagram Chris_W drew is correct. It is the rapid deflection back and forth, the "weaving," if you will, of the chain through the derailleur that causes resistance. When each link of the chain swings around one pulley and then immediately around the next, there is more of an arc on the pins to travel through, and therefore more resistance. The opposite effect can be felt on single speeds. A single speed bike in the same gear ratio as a multispeed bike (especially one with a short cage RD) will feel much smoother. I believe Sheldon Brown noted that it was even less tiring... I think it feels different, but the difference you can feel in an external bottom bracket versus an internal one is greater, and that has been measured to have an insignificant effect.

Chris_W is also correct that lateral deflection makes a much larger difference than does the vertical, but again, I doubt it makes any real difference other than "feel".

I just thought it was interesting.