Quote 16
(Odysseus:) 'I know well how to handle the polished bow, and would be first to strike any man with an arrow aimed at a company of hostile men … But I will say that I stand far out ahead of all others such as are living mortals now and feed on the earth. Only I will not set myself against men of the generations before, not with Herakles nor Eurytos of Oichalia, who set themselves against the immortals with the bow, and therefore great Eurytos died suddenly nor came to an old age in his own mansions, since Apollo in anger against him killed him, because he had challenged Apollo in archery.' (8.215-228)
It looks like Odysseus has learned his lesson since boasting to the Cyclops—at least, part of his lesson. He's still claiming to be the best archer living, but key word living. He's going to claim that he's the best archer ever or anything. Because that would just be bragging.
Quote 17
(Odysseus, in his tale:) ‘Cyclops, in the end it was no weak man’s companions you were to eat by violence and force in your hollow cave, and your evil deeds were to catch up with you, and be too strong for you, hard one, who dared to eat your own guests in your own house, so Zeus and the rest of the gods have punished you.’ (9.475-479)
Translation: "How do you like me now?"
Quote 18
(Odysseus:) 'So they spoke, but could not persuade the great heart in me, but once again in the anger of my heart I cried to him: "Cyclops, if any mortal man ever asks you who it was that inflicted upon your eye this shameful blinding, tell him that you were blinded by Odysseus, sacker of cities. Laertes is his father, and he makes his home on Ithaka." (9.500-505)
Odysseus managed to get (most) of his men off of the Cyclopes' island, but he just can't let a good thing be. He had to go and open his big mouth, practically giving Polyphemos his address and Facebook password. Smooth move, big guy.