Syracuse Arms. Co. has a very interesting history. But rather than retyping all of the available information, google Syracuse Forging and Gun Co. and Syracuse Arms Co. It could have been used as a stage coach gun, it could also been used to guard the corner liquor store which is most likely it's use.
However, who ever decided to cut the barrels down destroyed a very nice collectible shotgun. Plain Jane grade Syracuse shotguns in good condition will bring up to a thousand dollars. Yours, if in excellent condition ,maybe a couple of hundred, if in poor condition a whole lot less, Sorry about that