This instrument was used on NSW railways but its exact application is unknown.
If a cylindrical object is placed between its jaws and the central slider is brought up to touch it, the scale reads directly the diameter.
Simple trigonometry (left as an exercise for the reader ;-)) shows that this will apply when the jaw angle is 38.9 degrees (2arcsin(1/3)). If the angle were 60 degrees it would read the radius directly, and calipers are available from several manufacturers that do this, although they enclose the object from both sides and are mainly used to measure 3-bladed rotary cutters.
The vernier scale allows a reading precision of 0.001 inches.
Carter Tools was a Sydney manufacturer of hand tools, mostly for woodworking, in the 1950s and 1960s. This device seems out of character with respect to their other products and has been incorrectly described elsewhere as a 'centre finder' or 'vernier circumference finder'.