The Bow is a ranged weapon that can be found near the door to the 3rd Floor Corridor of Derceto. It can be seen leaning against the shelf there. It is unable to be used as a weapon until the protagonist has obtained the Arrows near the statue in the Forest Room.
The Bow is used in getting through to the Library entrance because the Bow and Arrows are capable of eliminating one of the possessed paintings, which is that of an Indian. The protagonist must fire at the portrait which is on the far side of the 2nd Floor Corridor.
Note: You actually don't need the bow to shoot the arrows. Simply throwing a lone arrow at the Indian painting should do the trick.