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Bilbo was scared! He had never had his feet off the ground before and when he found himself in the air, he held on too tightly.
bilbo has altophobia basically means he has a fear of heights so when he realized how high he was he got scared and probably naturaly grabbed tighter and bilbo is much smaller than the eagle which it felt like a pinch to the eagle evenm if it was bilbos whole hand another word for dear of hights is acrophobia :)
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