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Eurymachus was one of the men who hoped to be chosen as Penelope's husband. It was thought that Odysseus was dead, and "suitors" from all over Greece gathered to offer themselves as husband. Of course, it didn't hurt that whoever married Penelope would become king of Ithaca.
Here is what the eNotes study says about him:
Son of Polybus, Eurymachus is described as the "leading candidate" for Penelope's hand (XV.17-18). His name means "wide-fighting."
Eurymachus is shown to be arrogant, disrespectful, hypocritical, cowardly, and abusive. He is the second of the suitors to die by Odysseus's hand. Odysseus's words to him, after Eurymachus offers to make good on the damages the suitors have done to his household in his absence, are virtually the same as Achilles's words in response to Agamemnon's offer of a ransom for Briseis in Book 9 of the Iliad.
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