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King Aeolus gives Odysseus a bag of winds to help him sail safely back to Ithaca. Odysseus' men open the bag, thinking it contains treasure, which causes a storm to blow their ship off course.
Aeolus refuses to help Odysseus a second time because he believes that the gods must be against Odysseus, given his bad luck at sea. Aeolus fears that if he were to help Odysseus again, he would incur the gods' wrath and bring disaster upon himself.
Odysseus called out to Polyphemus that he was the one who blinded him as a tactical move to boast and reveal his identity. This was a strategic decision by Odysseus to show his cunning and attempt to gain favor with the Cyclops. However, it ultimately led to further conflict and challenges for Odysseus and his crew.
Eurymachus is one of the suitors vying for Penelope's hand in marriage while Odysseus is away. He is shown as a manipulative and deceitful character, trying to gain favor with Penelope and plotting against Odysseus. Eurymachus is part of the group that takes advantage of Odysseus's absence by feasting and trying to win over Penelope.
Odysseus wants to stay in Polyphemus' cave in order to receive hospitality and offerings as is customary in Greek culture. He hopes to gain favor with the Cyclops and secure his safety while trapped in the cave.
He tested the Herder by Pretending to be a beggar that has been worn down. He asks of the king (himself) and how he was a great warrior. The Herder did not have any idea that the king was in front of him the whole time.
King Aeolus gave Odysseus a bag of winds.
King Aeolus
As far as I know I don't think Aeolus is a king, though I may be wrong. He is definitely however a God. He the god of the winds. What he gives Odysseus in the bag is all the 'wrong' winds that are blowing the opposite way to Ithaca. By giving Odysseus the bag of winds, Aeolus is making it easier for Odysseus to get home quicker.
Odysseus remained with Aeolus for a month out of hospitality. Further, Aeolus wished to know about the war of Troy, and asked Odysseus many questions on the subject.
It gets loosed by the curious deceitfull shipmates of Odysseus while he slept.
Aeolus thought that the gods hated odysseus and banished them from his island
Oddysseus met Aeolus on the island of Aeolia
King Aeolus gives Odysseus a bag of winds to help him return to Ithaca. However, Odysseus' crew opens the bag, thinking it contains treasure, and the winds escape, blowing the ships off course and causing a storm that drives them back to Aeolus' island. This betrayal leads Aeolus to believe that Odysseus is cursed and refuses to help him further.
Aeolus
Odysseus regales Aeolus with stories of the war at Troy for a month, before leaving. When he returns, Odysseus begs Aeolus for more help, but he is denied.
He does not wish to incur the god's wrath. He can tell that the gods have cursed Odysseus. Also, Odysseus has already wasted his first gift.
Aeolus bottled up all winds but the west winds and put them in an ox-hide bag for Odysseus.