Swing Tailstock or Jewelling Tool:
Originally designed for rapidly opening holes in clock plates to receive jewels the unit was developed to enable it to accurately finish bore holes to receive parts that might vary slightly in diameter from piece to piece.
The aim of the device was to ensure the highest possible accuracy by holding, between a gauge plate and gauging finger at the top of the unit, the actual jewel (or lens, etc.) that was going to be fitted into the bored hole. If the setting instructions were followed the bored hole would be exactly the same size as the part held by the gauging device.  Stark Watchmakers' Lathe Page 1