Steve! You are the winner (and by that I mean you are the smart one this week)...It is a tooth extractor. Believe it or not the museum has more than one tooth extractor.
See this great illustration (fresh from Wikipedia) on how the extractor or "dental key"was used. Just as Steve said in his comments, it was inserted into the mouth, the hook was tightened over the tooth. Then twist and pull! Voila! Tooth! And maybe some jaw fractures, broken teeth, tissue damage. But hey, that toothache is gone. Right?
The dental key was invented in France, probably around 1730. It saw many improvements in the 18th century and 19th century, including interchangeable keys for the handle (to fit varying size teeth). However, it was replaced in the late 19th century by the forceps. Below is a crude version of dental forceps.
Most of the serious work of the 19th century dentist involved pulling teeth, although filling teeth will gold, silver and amalgam (a mixture) began as early as 1800.