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The peace-offering (like the burnt-offering, Leviticus 1:3, and the Minchah, Leviticus 2:1) is here spoken of as if it was familiarly known before the giving of the Law. “Peace-offering” seems preferable to “thank-offering,” which occurs in several places in the margin of our Bible. “thank-offering” appears to be the right name for a subordinate class of peace-offering.
Kill it at the door ... - See Leviticus 1:3. Tradition says that the peace-offerings might be killed in any part of the court.
“The fat that covereth the inwards” refers to the caul or transparent membrane which has upon it a network of fatty tissue: “the fat upon the inwards” refers to the small lumps of suet found upon the intestines of healthy animals.
The caul above the liver - Probably the membrane covering the upper part of the liver.
Upon the burnt sacrifice - Upon the ashes of the continual burnt-offering Exodus 29:38, in accordance with Leviticus 6:12.
A lamb - A sheep. The word signifies a full-grown sheep, in its prime.
See Leviticus 1:4-3.1.5 notes.
The whole rump - The whole fat tail: i. e., the tail of the kind of sheep well known in the East, and often weighing 15 lbs. and even as much as 50 lbs., when the sheep has been increased by artificial fattening.
Burn it - See Leviticus 1:9 note.
See Leviticus 1:10 note. Birds were not accepted as peace-offerings, most probably because they were, by themselves, insufficient to make up a sacrificial meal.
Rather, as food of an offering made by fire for a sweet savour, shall all the fat be for Yahweh. Our bodily taste and smell furnish figures of the satisfaction with which the Lord accepts the appointed symbols of the true worship of the heart. All that was sent up in the fire of the altar, including the parts of the sin-offering Leviticus 4:31, as well as the burnt-offering (Leviticus 1:9, etc.), was accepted for “a sweet savour”: but the word food may here have a special fitness in its application to the peace-offering, which served for food also to the priests and the offerer, and so symbolized communion between the Lord, His ministers, and His worshippers.
The fat is the Lord’s - The significance of this appears to consist in the fact that its proper development in the animal is, in general, a mark of perfection.
Blood - See Leviticus 17:11 note.
Throughout all your dwellings - The suet was neither to be eaten in sacrificial meals in the sanctuary, nor in ordinary meals in private houses.
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Barnes, Albert. "Commentary on Leviticus 3". "Barnes' Notes on the Whole Bible". https://www.studylight.org/
the Second Week of Advent