When they had made fast the running gear all along the black ship, then they set up the mixing bowls, filling them brimful with wine, and poured to the gods immortal and everlasting but beyond all other gods they poured to Zeus’ gray-eyed daughter. (2.430-433)
Pouring out perfectly good wine (and burning perfectly good ram's thighs) definitely sounds like a "sacrifice": it's a good way to convince the gods that you take them seriously.
Then in turn the gray-eyed goddess Athene answered him: ‘Telemachos, some of it you yourself will see in your own heart, and some the divinity will put in your mind. I do not think you could have been born and reared without the gods’ will.’ (3.25-28)
Athene tells Telemachos to have faith in himself and in the gods since they have always favored him. She herself, loving Telemachos for Odysseus’s sake, gives him the words and courage to speak eloquently to Nestor.
They came to Pylos, Neleus’ strong-founded citadel, where the people on the shore of the sea were making sacrifice of bulls who were all black to the dark-haired Earthshaker. There were nine settlements of them, and in each five hundred holdings, and from each of these nine bulls were provided. (3.4-8)
The residents show their piety by holding ritual sacrifices to their patron god, Poseidon. Their piety implicitly renders Nestor, ruler of this land, a trustworthy friend for Telemachos.
(Athene:) 'Hear us, Poseidon, who circle the earth, and do not begrudge us the accomplishment of all these actions for which we pray you. First of all to Nestor and to his sons grant glory, and then on all the rest of the Pylians besides confer gracious recompense in return for this grand hecatomb, and yet again grant that Telemachos and I go back with that business done for which we came this way in our black ship.' (3.55-61)
Weird. It's not just the humans who pray to the gods; gods also pray to other gods. Apparently there's a pretty strict hierarchy on Mount Olympus… but being Zeus' daughter does give you some perks.
(Nestor:) 'Act quickly now, dear children, and do me this favor, so that I may propitiate first of all the gods, Athene, who came plainly to me at our happy feasting in the god's honor. Come then, let one man go to the field for a cow, so that she may come with all speed, and let one of the oxherds be driving her, and one go down to the black ship of great-hearted Telemachos, and bring back all his companions, leaving only two beside her, and yet another go tell the worker in gold Laerkes to come, so that he can cover the cow's horns with gold. You others stay here all together in a group but tell the serving women who are in the house to prepare a glorious dinner, and set chairs and firewood in readiness, and fetch bright water.' (3.418-429)
Want your sacrifice to net you bonus points? Cover its horns with gold, first. We're getting the idea that pleasing the gods means performing extravagant wasteful actions—and, bear with us for a minute, but wouldn't people who have the means to perform extravagant wasteful actions have a leg up in the first place?
(Proteus, in Menelaos' tale:) '"But you should have made grand sacrifices to Zeus and the other immortal gods, and so gone on board, so most quickly to reach your own country, sailing over the wine-blue water. It is not your destiny now to see your own people and come back to your strong-founded house and to the land of your fathers, until you have gone back once again to the water of Egypt, the sky-fallen river, and there have accomplished holy hecatombs in honor of all the immortal gods who hold wide heaven. Then the gods will grant you that journey that you so long for."' (4.472-480)
Coulda, shoulda, woulda. We're pretty sure that Odysseus will do a lot of sacrificing before he steps one foot on a boat again.
'[…] and Aias would have escaped his doom, though Athene hated him, had he not gone wildly mad and tossed out a word of defiance; for he said that in despite of the gods he escaped the great gulf of the sea, and Poseidon heard him, loudly vaunting, and at once with his ponderous hands catching up the trident he drove it against the Gyrean rock, and split a piece off it, and part of it stayed where it was, but a splinter crashed in the water, and this was where Aias had been perched when he raved so madly. It carried him down to the depths of the endless and tossing main sea. So Aias died, when he had swallowed down the salt water.' (4.502-511)
Now for something completely different. We see a lot of examples of piety in the Odyssey, and here's an example of impiety. It buys you a one way ticket to certain, immediate death.
(Odysseus:) 'Hear me, my lord, whoever you are. I come in great need to you, a fugitive from the sea and the curse of Poseidon; even for immortal gods that man has a claim on their mercy who comes to them as a wandering man, in the way that I now come to your current and to your knees after much suffering. Pity me then, my lord. I call myself your supplicant.' He spoke, and the river stayed its current, stopped the waves breaking, and made all quiet in front of him and let him get safely into the outlet of the river.' (5.445-454)
We get the feeling that praying to an unknown river god is sort of like begging your car to start or your computer not to crash when you have six pages full of unsaved work. And we really wish it worked. (Maybe if we poured some Diet Dr. Pepper on the floor?)
(Polyphemos, in Odysseus' tale:) '"Stranger, you are a simple fool, or come from far off, when you tell me to avoid the wrath of the gods or fear them. The Cyclopes do not concern themselves over Zeus of the aegis, nor any of the rest of the blessed gods, since we are far better than they […]."' (9.273-287)
Not all divinities live on Mount Olympus. You have to be careful in The Odyssey, because there's always that chance that you're inadvertently ticking off some immortal with powerful connections—like a close family tie to the god of the sea.
(Odysseus:) '[…] for me alone my strong-greaved companions excepted the ram when the sheep were sheared, and I sacrificed him on the sands to Zeus, dark-clouded son of Kronos, lord over all, and burned him the thighs; but he was not moved by my offerings, but still was pondering on a way how all my strong-benched ships should be destroyed and all my eager companions.' (9.550-555)
We have to ask: how do you know that a god isn't moved by your offering? And if he's not, do you get to eat it yourself?
(Teiresias, in Odysseus’ tale:) ‘“But after you have killed these suitors in your palace, either by treachery, or openly with the sharp bronze, then you must take up your well-shaped oar and go on a journey until you come where there are men living who know nothing of the sea, and who eat food that is not mixed with salt, who never have known ships whose cheeks are painted purple, who never have known-well-shaped oars, which act for ships as wings do. And I will tell you a very clear proof, and you cannot miss it. When, as you walk, some other wayfarer happens to meet you, and says you carry a winnow-fan on your bright shoulder, then you must plant your well-shaped oar in the ground, and render ceremonious sacrifice to the lord Poseidon, one ram and one bull, and a mounter of sows, a boar pig, and make your way home again and render holy hecatombs to the immortal gods who hold the wide heaven, all of them in order. Death will come to you from the sea, in some altogether unwarlike way, and it will end you in the ebbing time of a sleek old age. Your people about you will be prosperous. All this is true that I tell you.”’ (11.119-137)
Here, Teiresias tells Odysseus about his ultimate fate – which will happen after the end of the Odyssey. How does this knowledge of Odysseus’ eventual death affect the mood at the end of the poem?
(Odysseus:) 'Then I went away along the island in order to pray to the gods, if any of them might show me some course to sail on, but when, crossing the isle, I had left my companions behind, I washed my hands, where there was a place sheltered from the wind, and prayed to all the gods whose hold is Olympos; but what they did was to shed a sweet sleep on my eyelids […].' (12.333-338)
Check out how Odysseus washes his hands first. This is a cool little detail about everyday religion in Ancient Greece—and it probably says something about just how dirty they were.
(Odysseus:) ‘You dogs, you never thought I would any more come back from the land of Troy, and because of that you despoiled my household, and forcibly took my serving women to sleep beside you, and sought to win my wife while I was still alive, fearing neither the immortal gods who hold the wide heaven, nor any resentment sprung from men to be yours in the future. Now upon you all the terms of destruction are fastened.’ (22.35-41)
Odysseus cites the suitors’ crime as one not only of incivility, but of impiety as well.
(Odysseus:) 'Keep your joy in your heart, old dame; stop, do not raise up the cry. It is not piety to glory so over slain men. These were destroyed by the doom of the gods and their own hard actions […].' (22.411-413)
Piety isn't just about sacrificing to the gods—it's also about how you treat your fellow men. It's not quite "do unto others," but Odysseus is reminding the overly enthusiastic woman that these men are dead because the gods wanted it to be that way. Making a fuss about it just isn't respectful.
(Eurymachos:) ‘[…] in any case we fear no one, and surely not Telemachos, for all he is so eloquent. Nor do we care for any prophecy, which you, old sir, may tell us, which will not happen, and will make you even more hated.’
Eurymachos and the suitors disrespect the gods by dismissing Telemachos, who is favored by Athene, and Halitherses, who speaks the gods’ will. By laughing off the words of those through whom the gods speak, they are placing themselves in opposition to divine law. So they will be punished.