In reply to cymjt:
You need two figures.
1. Chainring capacity, this is the number of teeth on the largest chainring minus the number of teeth on the smallest chainring.
2. Cassette capacity, this is the number of teeth on the largest sprocket/cog minus the number of teeth on the smallest sprocket/cog.
There's a third figure which is the sum of these two which is drivetrain capacity.
So having done that for Shimano the following holds:
SS - Short Cage Road Double - Maximum Cassette Cog is 27 and Total Capacity is 29
GS - Medium Cage MTB/Road Triple - Maximum Cassette Cog is 34(MTB)/27(Road) and Total Capacity is 33(MTB)/37(Road)
SGS - Long MTB - Maximum Cassette Cog is 34 and Total Capacity is 45
Your proposed new cassette has a capacity of 18. A Tiagra triple chainset has 52, 39, 32 chainrings so has a capacity of 20. Thus you have a drivetrain capacity of 38
Now you can usually get away with being a bit over the stated capacity but it can be a bit tight. From all the above I'd say that you can use a medium cage rear mech.