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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.
Eurymachus is one of the suitors in the Odyssey who has taken over Odysseus's home. In Book II, after Zeus sends down two eagles as a sign that Telemachus's warning to the suitors to leave Odysseus's palace should be obeyed, Eurymachus replies, "I’m a better hand than you at reading portents. / Flocks of birds go fluttering under the sun’s rays, / not all are fraught with meaning." Eurymachus blithely avoids listening to the signs of the gods, and he says that Odysseus is dead and that Penelope should marry one of the suitors. He also discounts Telemachus as a threat. He constantly disobeys the signs of the gods and flouts the Greek laws of hospitality.
For a while, Eurymachus (along with Antinous) seems like the suitor who is most likely to marry Penelope, as he provides the best gifts, and even her brothers and father urge her to marry him. When Odysseus returns home, disguised as a beggar, Eurymachus, not following the Greek laws about treating guests with graciousness, mocks him. Odysseus slays Antinous first, and Eurymachus, trying to be crafty, blames Antinous for all the bad deeds the suitors have carried out. Odysseus kills Eurymachus with an arrow that pierces his liver.
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