What is the significance of "I am become a name" in "Ulysses"?

Quick answer:

The significance of the line "I am become a name" is that it shows us Ulysses' realization that a good reputation is not enough. Ulysses is one of the most famous characters in ancient mythology, yet he knows that he cannot rest on his laurels. To live up to his name, he must embark on yet more heroic journeys and adventures.

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One could be forgiven for allowing Ulysses—known in Greek mythology as Odysseus—a well-earned rest. After all, this extraordinary man, this living legend, has crammed more heroic adventures into his life than most people could possibly conceive.

But Ulysses isn't like us; he lives in a society in which heroes constantly need to live up to their heroic reputations. Even though Ulysses is getting on a bit, and should ideally be putting his feet up at this time of life, he feels the irresistible pull of the wanderlust that rages deep within his soul.

Though he has "become a name," that's not all he is. A hero is only as good as his most recent heroic deed. A Ulysses who sits at home doing nothing, a Ulysses who rests on his laurels, is a Ulysses in name only; such a man no longer truly embodies what it means to be Ulysses.

This explains Ulysses's determination to gather together a ship's crew and embark upon yet another action-packed epic voyage to some strange, exotic land or other. It explains why Ulysses, despite his advanced years, is so restless; why, after all this time, he still yearns "To strive, to seek, to find, and not to yield."

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