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 70 from 1885. 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, coin-silver case, which makes a sidewinder. I have seen photos of this in hunter cases (a hunter case has a hinged cover to protect the dial glass), but this ain't one of 'em.
This particular grade is a B.W. Raymond, which was named after the company's founder. Raymonds are high-end Elgins. In this case, it's 15 jewels where others from this period are 7-jewel models, and it's adjusted for 2 positions (meaning it's accurate in two different positions—probably pendant up and dial up, but I can't be sure). That makes it more valuable and useful for people who depended on accurate time, like railroad conductors and engineers. While this wasn't "railroad grade" as it came to be defined later, it was an early version of it. I would not be surprised if it had been carried by someone who worked on the rails.
The case is a Fahys Coin No. 1 case.
Currently not working. If I ever get a watch repair budget, this would be high on the list.
Acquired: 2019
Circa: 1887
Current condition: doesn't work, don't know why