Well, I was a little surprised at the use of the term, but Al reiterated it so I take it at face value.
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Pivot Gun
A pivot gun was a large gun mounted on a pivot or revolving carriage, so as to turn in any direction. By 1812 the long gun on a pivot had reached the height in popularity in America, which it maintained for sixty years, until it was replaced by the turret mount [which in anticipated]. The pivot gun mount typically consisted of a metal ring, or 'circle' of from 9' to 12' in diameter, on deck and brought level athwartships by a wooden foundation. This circle was usually of iron, though copper and brass could be employed. The section of the circle was a shallow "U" shape, hollow side up, about 5 1/2" wide and 1" thick. The inside and outside rims of the top were raised 1/2" and were about 1/2" wide, creating a track in which the rollers traveled.
The rollers were on the bottom of two horizontal timbers, 8" to 12" square, called 'skids,' secured by three or more blocks, or 'chocks,' and bolted. The skids were parallel and usually a few feet apart. On the top inside edge of each there was a rabbet running the full length of the skid. In this the bottom of the gun mount could slide. The skids were pivoted at the middle, or thereabouts, by a heavy pivot bolt, or pin, which passed through the center chock of the skids and thence through deck and a heavy timber plate in the deck and was often heavily bushed in the skid-chock, since the strains of recoil were largely concentrated on this structure.
The gun mount consisted, as a rule, of the standard broadside carriage without trucks, the bottom of the side brackets of which rested in the rabbets on the upper and inner side of the skids. Sometimes there were rollers on the underside of the brackets, or the trucks were retained and traveled in the grooves in the skids. The gun was trained by prying the skids around by means of handspikes. Recoil was controlled by breechings - heavy rope secured to the breech of the gun and fastened either to ringbolts in the deck about the gun, or on neighboring bulwark stanchions. Small guns had breechings secured to the skids, but this put a greater strain on the pivot bolt than was desirable, so when the gun was brought to bear on a target the breechings were commonly hooked onto ringbolts in deck and rail. Neither gun nor mount was particularly suitable for firing on fast-moving targets."
- from Globalsecurity.org
Now, elsewhere I've read that the ring could also be used as a calibrated sector on which the bearing of the gun relative to the ships centerline could be accurately gauged, which in turn aided aiming the gun with mathematics.
As for the Cap'n, yes I recall a photograph that I would very much like to find, of a crew manning a .50 caliber water cooled machine gun on the deck of a super dread US battleship. The sweep if you will, was a very large and elaborate cage structure all around the mount. Which became evident of it's intent seeing the whole of the bridge structure above and behind it.