Welcome to the forum. Looks like you have a prize there. You can go to
Horn-u-copia and scroll down to the Manufacturer's listings, then to the Pepper thread. Lots of info there. I see a Picture of a Pepper Surprise, no 2850 in the "exploring serial numbers" postings. There is a clickable serial number list there also, but not complete. There is another Surprise picture, repy 60, 55454.
On the 1902 Pepper brands thread "surprise- reputable foreign maker" from the J.W. Pepper Musical times and Band Journal.
Other info on reply #4. Surprise 1 water key, $13.50 ( I assume), $17.75 -2 water keys. Silver, GWB finish. "for comparison purposes, a Conn Bb slide cornet was being advertised i the same issues as being used, for $40."
On the Pepper cornets thread, I see a Surprise serial 31726 (c. 1984-1902) , another serial 70277 (c. 1919) (reply #8), another on reply #11, no serial , but notation "made in Austria" 12.5 in. long, another, reply #23, a Surprise serial #62098 ( c. 1900s?)
Hope this helps a bit. Happy Hunting!
