I'm not sure if it was ever done in the Civil War, but during the Napoleonic Wars leg amputations were often done with the patient sitting up as the Zouaves are doing. This is for ergonomic reasons - it's easier for the surgeon to operate the knife in this position. The surgeon holds the the blade so the edge is facing his body and cuts in a circular motion from underneath, separating first the skin and then the muscle. Then the bone saw saws through the bone.
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Colorized but smaller:
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