This is a more accurate and more time efficient manner in which to cut jump rings. A large number of rings can be cut in a relatively short amount of time. Wind the rings around a mandrel (I like to use wood for this method of cutting rings).
Then place a low-tack masking tape along the coil. The low-tack tape serves several purposes: it prevents residue from being deposited on the rings while being cut, and it serves as a stabilizer to the rings during the cutting.
Using the wooden dowel as a stabilizer, gently cut the rings and allow the saw to cut into the dowel just slightly.