Well, yes, it does sound funny, but it is surprising how well it works.
The initial idea (for me) came from hearing of an Australian woman who was evacuating from a firestorm several years ago, and that's what she did.
I recalled it one day when I had to take several cats (including my sister's, who live with her, now) to the vet, and didn't have enough carriers. Apparently, not being able to see the landscaping whizzing by, they were extraordinarily calm and quiet.
However, I did learn there is a "best way" to do it, for anyone interested:
Place the pillowcase on a bed or floor, and open it with the end seam as the bottom, and gather all "sides" in a circle around the exposed bottom. Set cat calmly on the bottom, then quickly bring up the sides and close it so the cat is completely covered. Tie securely with a shoelace. (I've tried everything -- shoelaces are best.)
For those who want to just stuff an upset cat into an open bag with one hand on each, please refer to Blast's recent epic: "Hi, my name is Blast and I'm bleeding" at
http://forums.equipped.org/ubbthreads.php?ubb=showflat&Number=106205#Post106205This method also works well for snakes, but not for chickens.
Sue