What is the significance of the quarrel in book 1 of Homer's Iliad?

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The quarrel between Agamemnon and Achilles is absolutely crucial to the action of book 1 of the Iliad and provides a catalyst for much of the action to follow.

Proud Achilles, the finest of the Achaeans' warriors, is furious at King Agamemnon's appropriation of his sex-slave Briseis. Under the circumstances, Agamemnon had little choice. He had to return his own sex slave Chryseis back to her father—a priest of Apollo—in order to lift the terrible plague sent down as punishment by Apollo himself. As Achilles's superior, he was perfectly entitled to take Briseis for himself.

But that does nothing to assuage Achilles's almighty wrath. As well as being angry, he's also deeply humiliated by Agamemnon's actions, so much so that he retreats to his tent, where he sulks like a big baby. For good measure, he even prays to his mother, the sea-nymph Thetis, to ask almighty Zeus to punish the Achaeans. Yes, that's right; Achilles actually prays for his own side to be punished, and all because Agamemnon took his sex slave away from him.

Without their finest warrior, the Achaeans are getting well and truly hammered by the Trojans, who are now starting to gain the upper hand in this epic battle. Agamemnon tries desperately to placate Achilles, offering him valuable treasure to lure him out of his tent. But Achilles is not interested. He will only reenter the fray when his bosom buddy Patroclus is killed in battle by Hector, breaker of horses.

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In the Iliad's first book, the conflict between Agamemnon and Achilles (and the result of that conflict) has tremendous implications over the course of the war.

The poem opens after Agamemnon has taken Chryseis (the daughter of Chryses, a priest of Apollo) as a war captive, with her father offering a ransom for Chryseis's return. When Agamemnon refuses to accept said ransom, he angers Apollo (which results in no small suffering for the Greek forces). Ultimately, it is determined that Agamemnon must release Chryseis to satisfy Apollo, but Agamemnon demands that, if he is to release Chryseis, he must be compensated with another man's captive. When this brings him into conflict with Achilles, Agamemnon determines to take Achilles's own captive, Bryseis. This rift between Achilles and Agamemnon has critical implications for the remainder of the poem.

Outraged, Achilles responds by removing himself from the war, while also appealing to his mother (the nymph, Thetis). Thetis herself appeals to Zeus, asking him to support Troy to punish this slight against her son. In this way, Achilles's complaint against Agamemnon actually has a cosmic dimension to it as well, through Thetis and her intervention with Zeus, most powerful of the gods.

At the same time, remember that Achilles is the greatest warrior among Agamemnon's forces. From that perspective, his removal from battle represents a significant loss to the Greeks, while the war is still ongoing. Recognizing this, the Greeks will later try to pacify Achilles, but it is not until the death of his friend...

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Patroclus at the hands of Hector that Achilles, enraged, returns to the battlefield.

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The quarrel between Achilles and Agamemnon in Book 1 of Homer's Iliad sets in motion a chain of events that will affect the tenth and final year of the legendary Trojan War.

In the first nine years of the war, little had taken place because the Trojans had realized in the early days of the war that they were no match for Achilles on the open battlefield.

Once Achilles leaves the fighting, though, this gives the Trojans the courage to leave the safety of their city walls and come out to fight. Hector inflicts serious damage on the Greek fortifications on Troy's shores and even threatens to set fire to the Greek fleet. The threats to the Greek ships causes Achilles' best friend Patroclus to want to enter battle. In Iliad 16, Patroclus' death, at the hands of Hector, causes Achilles to return to the fighting and kill Hector (Iliad 22). Achilles then holds funeral games for Patroclus (Iliad 23) and ransoms back Hector's body to Priam (Iliad 24).

We should note, however, that the Trojan War does not end at this point. We should also note that Hector's death did not mean the defeat of the Trojans. The Trojans would get reinforcements. We should also keep in mind that Achilles himself did not live to see the fall of Troy. He is killed later in the tenth year by Paris/Alexander.

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