There's no sorting it out. I referred to a ball-pein hammer the other day, a tool sometimes used to peen over the end of a rivet, and got corrected to the latter spelling. As it happens, both are acceptable and I learned the "ei" spelling as a child.
Cross-pein hammers are seen in metalwork -- a blacksmith might use cross-pein, straight-pein and diagonal-pein hammers -- and "pinhead" hammers are not unknown, with the pein side tapering down to small cylindrical end. "Pin" and "pein" (or "peen") seem to share a root word, with a meaning something like "peg." But wait! A woodworker might use a Warrington-pattern hammer, a sort of modified cross-pein hammer with a longer pein end, but among them you're about as likely to hear it described as a cross-pane hammer, with the explanation that the long, truncated-wedge pein is there to drive the nails that hold the pieces of wood that comprise the muntins (aka glazing bars) and, I suppose, the mullions of a multipane window without risking striking the glass.
What's the real story? I don't know. "Pane" sounds suspiciously like folk etymology, and may suffer from the same sort of cross-Atlantic phonetic shifts that have an American cabinetmaker carefully making a rabbet along a board while his British cousin forms a rebate, and yet they're doing the exact same thing.
Whichever term you use, let the weight of the hammer head work for you and don't "choke up" on the handle -- use the whole thing and if you need less force, go get a smaller hammer. And by "smaller hammer," I include one of my favorites, the tack-hammer edition of the Warrington-pattern known as a "telephone hammer." Why? Well, you see, in the day of wooden wall-mounted telephones, it was easier to carry them around in knocked-down form, and tap in a few brads to hold the body of the thing together as part of the installation process....
1 month ago