Although several people finally identified the previous "Whatchamacallit", Stan Rowe was kind enough to send me some pages from a book published in 1960 titled "The Story of Pope's Barrels" by Ray M Smith. Harry Pope was well known for his rifle barrels and shooting aids. This photo shows the Pope Machine Rest in use at the breech of the old rifle. There is another piece (which I do not have) which clamps to the barrel at the muzzle end. Thus, it is easy to return the rifle to its original position while sighting it in. Thanks to all who enabled me to determine the use, and origin, of this item.