exact item shown, from an estate, not sure exactly what it was for, it has a tapered section whose diameter is .2980 by the threaded part (threads are 1/4-20), and .3595 at the ridge near the center.  on the other side of the ridge it is .359" all the way to the MT holder.  This could have been a specialty tool for working on small parts with a through hole, or it could hold a selection of buffing/polishing/deburring wheels, or it could be useful for making lamp bases on your wood lathe since 3/8 inch is a typical lamp rod size.