A sacrifice is a ritual, in which something is "consumed", i.e. killed, destroyed, eaten or given away. The sacrificed may refer to the giver's person or characteristics, e.g. may a farmer sacrifice cattle, or it may refer to the receiver, i.e. a sacrifice of wine for the god of wine. Or it may refer to a union of the giver and receiver. In all cases the ritual of sacrifice establishes a connection between giver and receiver. That, which is sacrificed, comes to represent both worlds, the present and the supernatural. Source: Gads Religionsleksikon, 1999.
A sacrifice does not need to be a personal offer, but this is often the case in Andersen's oeuvre. Indeed,
The little mermaid set out from her garden toward the whirlpools that raged in front of the witch's dwelling. She had never gone that way before. No flowers grew there, nor any seaweed. Bare and gray, the sands extended to the whirlpools, where like roaring mill wheels the waters whirled and snatched everything within their reach down to the bottom of the sea. Between these tumultuous whirlpools she had to thread her way to reach the witch's waters, and then for a long stretch the only trail lay through a hot seething mire, which the witch called her peat marsh.(...)
She reached a large muddy clearing in the forest, where big fat water snakes slithered about, showing their foul yellowish bellies. In the middle of this clearing was a house built of the bones of shipwrecked men, and there sat the sea witch, letting a toad eat out of her mouth just as we might feed sugar to a little canary bird. She called the ugly fat water snakes her little chickabiddies, and let them crawl and sprawl about on her spongy bosom.
"I know exactly what you want," said the sea witch. "It is very foolish of you, but just the same you shall have your way, for it will bring you to grief, my proud princess. You want to get rid of your fish tail and have two props instead, so that you can walk about like a human creature, and have the young Prince fall in love with you, and win him and an immortal soul besides." At this, the witch gave such a loud cackling laugh that the toad and the snakes were shaken to the ground, where they lay writhing.
"You are just in time," said the witch. "After the sun comes up tomorrow, a whole year would have to go by before I could be of any help to you. J shall compound you a draught, and before sunrise you must swim to the shore with it, seat yourself on dry land, and drink the draught down. Then your tail will divide and shrink until it becomes what the people on earth call a pair of shapely legs. But it will hurt; it will feel as if a sharp sword slashed through you. Everyone who sees you will say that you are the most graceful human being they have ever laid eyes on, for you will keep your gliding movement and no dancer will be able to tread as lightly as you. But every step you take will feel as if you were treading upon knife blades so sharp that blood must flow. I am willing to help you, but are you willing to suffer all this?"
"Yes," the little mermaid said in a trembling voice, as she thought of the Prince and of gaining a human soul.
"Remember!" said the witch. "Once you have taken a human form, you can never be a mermaid again. You can never come back through the waters to your sisters, or to your father's palace. And if you do not win the love of the Prince so completely that for your sake he forgets his father and mother, cleaves to you with his every thought and his whole heart, and lets the priest join your hands in marriage, then you will win no immortal soul. If he marries someone else, your heart will break on the very next morning, and you will become foam of the sea."
"I shall take that risk," said the little mermaid, but she turned as pale as death.
"Also, you will have to pay me," said the witch, "and it is no trifling price that I'm asking. You have the sweetest voice of anyone down here at the bottom of the sea, and while I don't doubt that you would like to captivate the Prince with it, you must give this voice to me. I will take the very best thing that you have, in return for my sovereign draught. I must pour my own blood in it to make the drink as sharp as a two-edged sword."
"But if you take my voice," said the little mermaid, "what will be left to me?"
"Your lovely form," the witch told her, "your gliding movements, and your eloquent eyes. With these you can easily enchant a human heart. Well, have you lost your courage? Stick out your little tongue and I shall cut it off. I'll have my price, and you shall have the potent draught."
"Go ahead," said the little mermaid.
The witch hung her caldron over the flames, to brew the draught. "Cleanliness is a good thing," she said, as she tied her snakes in a knot and scoured out the pot with them. Then she pricked herself in the chest and let her black blood splash into the caldron. Steam swirled up from it, in such ghastly shapes that anyone would have been terrified by them. The witch constantly threw new ingredients into the caldron, and it started to boil with a sound like that of a crocodile shedding tears. When the draught was ready at last, it looked as clear as the purest water.
"There's your draught," said the witch. And she cut off the tongue of the little mermaid, who now was dumb and could neither sing nor talk.
"If the polyps should pounce on you when you walk back through my wood," the witch said, "just spill a drop of this brew upon them and their tentacles will break in a thousand pieces." But there was no need of that, for the polyps curled up in terror as soon as they saw the bright draught. It glittered in the little mermaid's hand as if it were a shining star. So she soon traversed the forest, the marsh, and the place of raging whirlpools.