I'm no expert, but I thought Reminton's first shotgun was the 1889. For something that old, I'd think you would want to get an expert appraisal.
Trapper John
MOST, Old side by side shotguns aren't worth a whole lot, price is based on how bad someone wants one.
More likely a Remington 1882.
If it is, in fact, from 1862, it will be a percussion muzzleloader. If it breaks open to take a shotshell it is almost certainly an 1882 model, or one of it's immediate predecessors, the Whitmore Hammer Lifter.
http://www.remingtonsociety.com/rsa/journals/M1882
http://www.wisnersinc.com/additional_info/RemingtonDblID.htm
http://www.gunbroker.com/Auction/ViewItem.aspx?Item=155829154
Jeez Tom, that shotgun is almost as old as you!
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