I believe they were actually the National Watch Company, based in Elgin, Illinois, but incorporated the city into their official name at some point.
Elgin concentrated on middle-class watches: lots of 7 through 15-jewel jobs. Not the best, but you don't need 23 jewels to get the job done. This was my father's favorite brand, so I have a bunch of them.
This is my Elgin Grade 316. By the serial number, it was made in 1909. It is size 18, 15 jewels, pendant wind, lever set. It was designed for a hunter case but mine was fitted into an open-face, nickel-silver case, which makes a sidewinder. I believe this is a common configuration on 316s.
Very similar to my Grade 111, but this one has arabic numerals. Unfortunately it's unadjusted and not railroad grade. A big, hefty watch.
Acquired: 2019
Circa: 1909
Current condition: runs fine