Angela, I make probably 100+ k'scopes per year that require anywhere from 1 1/16" to 1 3/4" boring up to 10" deep. I do the same as Phil above, and would never try this on a drill press. It's a fair amount of work but if you turn the lathe down slow, a good 4 jaw chuck, and the bit in a drill chuck on the tail stock it can be done. Clear the chips often, and each species of wood behaves differently.
One bit of advice- bore out the center, rough turn it round and set aside to fully dry a while before finish turning and assembly. After it dries a while I always run the bit in it one more time, the wood almost always shrinks no matter how dry it was when you started and this will save a lot of trouble getting the mirrors in it. It almost always ends up oval shaped when it shrinks also, reboring solves that problem also.