"and pressed the whatever-the-end-is-called against the door"

 

Stethoscope
Public DomainStethoscope - Credit: Huji

The end is simply called the chestpiece.  It has two sides, a "bell" for picking up low frequency sounds, and a "diaphragm" for higher frequency sounds.

The stethoscope was invented by René Laennec in 1816, at a hospital in Paris. In 1852 George Cammann modified the design.  Rappaport and Sprague innovated the two-sided chestpiece in the 1940s.