Through Saddle
The old style for guitars
A steel string guitar bridge is said to have a "long" or "through" saddle if the saddle fits into a groove cut all the way through at the ends of the bridge:
Martin guitars from the 1920s to 1960s were made with this type of saddle.
Cutting the saddle all the way through at the ends causes the bridge to be a bit weaker, so the saddle is customarily was glued in place.
By contrast, this bridge has a "blind" saddle: