Hi -
Welcome to the forum! Glad to see you here.
Absolutely beautiful Hamiltons you have there.
In some ways it's not miraculous at all that your Hamilton keeps near-perfect time: Hamiltons of that era are the American equivalent of a Patek-Philipe watch today. Simply the best.
And if it was used regularly on railroad duty, then it was maintained and regulated on a set basis every month to ensure that it was truly of the accuracy needed for railroad use. That is often not known: part of the accuracy of railroad watchs is the fact that the users were regularly controlled and checked to make sure that their watches were properly taken care of, as doing something as silly as forgetting to wind it would literally kill people if the watch stopped and timing was off in the days of single-track connections. As a matter of fact, that is why there are railroad-quality watches: people died because a conductor didn't notice that his watch had lost time, resulting in a major collision.
But consider doing yourself a favor and do get it looked at: the expense of properly maintaining that watch will be rewarded by you be able to give to your grandchildren at one point and having them post at a future incarnation of WatchUSeek about how it keeps near-perfect time 200 years later...
Thank you for sharing those pictures as well. I'm a wristwatch guy, but the level of finishing and the sheer horological joy of those Hamiltons makes me want to get one simply to be able to take it out and look at it...
JohnF