Hi, Dianna. You will find that leaving the floor unfinished under the rug was common practice a hundred years ago. Perhaps thrift was part of it but it also may have been that the finishes would stay sticky long enough that the rug would get stuck to the floor. This would make cleaning difficult as the women would not be able to take the rugs outside and beat the dirt out of them. Aren't you glad that you live in the post vacuum cleaner era?
There are really two ways that you can approach this. By the way, a floor where some was covered and some was open to the effects of sunlight and age would still show a lot of variation over 100 years or so. Since this is a natural phemomena that has taken a hundred years to obtain, you might wish to just strip the outside edges and then finish the whole floor. If you see a guest looking at this, you can explain the cause and tell them that your house proudly wears this as a sign of it's longevity. Or you can sand the floor, finish, and not worry about it. The choice is yours.