What is the importance of Alcinous (Alkinoos) and Polyphemus (Polyphemos) in Book 9 of Homer's Odyssey?I can't seem to find it in the book..
In Book 9 of Homer's Odyssey, we find Odysseus beginning to tell his story to the people of Phaeacia. Odysseus has been with the Phaeacians for three books now (6-8), but he does not reveal his identity to his hosts until late in Book 8.
Alcinous is one of Odysseus' hosts. He is the king of Phaeacia. His wife is Arete and they have a daughter, Nausicaa, who is the first to discover Odysseus after he is shipwrecked on Phaeacia. So, I suppose Alcinous' importance in Odyssey 9 would be that he is perhaps the most politically powerful member of Odysseus' audience over the next four books (9-12), because he will be the one to give the order to his people to take Odysseus back to his native land of Ithaca.
As for Polyphemus, readers may know him better as the Cyclops who eats some of Odysseus' men. Obviously, he is important because he kills some of Odysseus' men, but he is also significant because he demonstrates extremely bad hospitality (Greek: xenia). Hospitality is surely one of the three most important themes in the Odyssey.
Polyphemus is also important because, near the end of Odyssey 9, Odysseus reveals his true name to him. At one point, he told the Cyclops that his name was Nobody. When Odysseus reveals his true identity to Polyphemus, the monster prays to his father Poseidon for revenge. Thus, much of what Odysseus suffers upon the seas is due to Polyphemus' prayer to Poseidon.