I think Ted Ernst is on the right track here... something is missing.
The U-shaped hoop looks like an early "Roller-Lever" type rim brake component. According to an old CTC [UK] publication, these brakes first appeared on a 1905 Sunbeam. They were then continuously produced in one configuration or another for many decades.
The one on this bike seems to have parts missing. A series of rod linkages would attach that "U-loop" to a seat post clamp and the brake pads were attached near the ends of the "U". The open ends of this "U" would be clamped to guides on the stays. This loop would slide forward when the pedal actuated the connecting rods. Then, when the mech drew forward, the pads would simply pivot inward upon contacting the bevelled rim to press flush against the rims brake flats...
Uh... at least that's how I imagine the Peugeot set would work.
Bob Hanson, Albuquerque, New Mexico, USA