I just wanted to add a little data to my original post. I did some math on the wire and it appears that each wire in a Cat 5 cable is 24 gauge or 0.0201 inches in diameter. I did find some that says that if the wire is really long runs, there may be cable out there that has 22 gauge wire.
Using 24 gauge, if you multiply that by 4, that is 0.0804 inches diameter which is roughly the diameter of a 14-16 gauge wire. So obviously this is more than needed for the 18-24 volts from the tranformer to the ring but it is a work-around for folks like me who just have a Cat 5 cable from their doorbell.
Thank you.