What sight made Bilbo's heart leap?
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The eagles make his heart leap....... His "eyes were seldom wrong" and he is the one who sees "the sudden gleam in the gloom"...a sight that made his heart leap." His announcement of the Eagles' approach is what enables the victory. As a final irony, a scene of birds flying overhead or a cry of "The Eagles" or "The Birds" is usually a symbol of doom. Here, Bilbo's cry of "The Eagles" is a herald of something positive though it may account for his own unfortunately injury.