How does the relationship between Virgil and Dante change throughout Inferno?
I would argue that the relationship between Dante and Virgil doesn't change that much over the course of The Inferno. When Virgil first appears in canto 1, he offers to help Dante, who has been stalled by the she-wolf blocking his path to the hill (Heaven). Virgil tells him that he knows another way; it involves traveling through all of the circles of Hell, but Virgil promises to be Dante's guide and companion.
As Virgil leads Dante through the vestibule of Hell in canto 3 and all of the nine circles that follow, he repeatedly prepares Dante for what he is about to see and protects him from those who question his presence (since he is not dead nor damned yet). Virgil tells Charon, for example, that it is not his place to question what God has deemed necessary. Dante faints repeatedly, overcome by his fear and disgust, and Virgil supports him and helps him move on to the next level.
Finally, in the last ring of the ninth circle, the poets see Satan and the three worst traitors of history—Judas, Brutus, and Cassius—each placed in one of Satan's three mouths. To exit Hell they must literally climb Satan's giant hairy body. Virgil actually carries Dante on his back, symbolizing not only that a sinner must overcome Satan and sin to get to Purgatory or Heaven but also that most sinners need assistance in their quest. As he has done all along, Virgil helps Dante complete his journey through Hell and allows him to move on to purgatory in the next part of The Divine Comedy.
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