What Is It?

These gadgets are a classic watch school project from the early 20th century.  They are the enamel sub-seconds parts from pocketwatch dials, on a brass base.  They can be spun like a top.  the top of the upper shaft is threaded and fit with a nut and the nut has a tapered body, facing down.

This tool is an aid in truing hairsprings.  A hairspring, with collet, can be held by the nut.  The taper centers it.  The white dial makes it easier to observe the spring, while turning the tool causing the spring to expand.  Hairsprings need to expand and contract evenly, and without their coils ever touching.