Original Remington-Rider Double Action Belt Revolver. 
This pattern Revolver was produced by the E. Remington & Sons of Ilion, NY, starting in 1863 and halted production in 1875 only producing between 3,000 to 5,000 of the .36 caliber revolver. The first 3,000 produced had a fluted cylinder and low serial numbers staying below 1000. The remainder of the 5,000 were produced in like manner of the New Model Army in 44. Caliber with a smooth surface cylinder and solid frame. The key difference between the Rider and the New Model was the Rider was a .36 caliber to the New model .44 and was Double Action. 
This particular example is a later produced model with the smooth cylinder and serial number #4588 located under the grips. A very beautiful “Grey” patina to it, no cracks in the complete wood grips and working action that may need some work. This gun has all the signs of a used firearm that saw plenty of use, and is rare piece of Remington firearms history!