In this chapter, the LORD commands how the priests and the altar are to be consecrated by animal sacrifice.
This chapter contains sacrifices of consecration, which in my opinion are very similar to sacrifices of atonement. The concept of atonement is to atone, or compensate, for some wrong committed in the past. The concept of consecration is to somehow purify or change an object/person so that it is now "holy", i.e. different or set apart from the common. Since holiness is always associated with the LORD, I believe it is also meant to express a sense of sanctity, that not only is it different from the "common", it is also pure. This is how inanimate objects like the altar are sanctified: obvious the altar has no sin to be atoned, yet it is still regarded as "common" until it is consecrated. The process of consecrating, then, must be a method of purifying and sanctifying the altar so that it is no longer common, it is now holy.
The priests, who are also consecrated, both have their sins atoned (because the sinful or impure cannot be holy) and their lives sanctified and purified in the same sense as the altar.
We definitely see an element of substitutionary atonement when the priests lay their hands on the animals' heads. This is a specific act of transference of guilt, which is a direct embodiment of substitutionary atonement, and another reason why I think consecration implies atonement.
As I previously noted, the priests are commanded to set blood on the horns of the altar. Later we will see more references to acts specifically performed on the horns of the altar. This shows that the horns are considered representative of the larger altar and what the altar itself represents.
Putting blood on the right ear, thumb and toe is reminiscent of the Passover (putting blood on the doorframes) and when Moses sprinkled blood on the people. On the one hand, it is an act of atonement like we see in the Passover, but on the other hand, it is a way of confirming and sealing a covenant (in this case) between the priests and God. I don't know the specific reason why they used the earlobe, thumb and toe, but there is a clear similarity to the Passover in how they are sealed with blood.
The anointing oil is a very specific oil mixture, which we will see the recipe later. This is a "sacred anointing oil" only used for religious purposes.
Also note that the priestly garments are only worn near the tent of meeting, within the courtyard (v. 4-9, 29-30). This confirms my hypothesis from the last chapter based on the usage of gold in creating the garments.
Emphasis on unleavened bread, even though this is not the Passover anymore. It is hard to explain this in the context of haste, which was true for the Passover. No other explanation given, but later Israelites considered leaven to be symbolic of sin or impurity.
The text here doesn't explain why the fat is burned on the altar but not the other parts of the bull. My guess is that it's because the fat is the highest caloric part of the animal and is considered the "best part", so burning the fat is a metaphor for giving the best part of the sacrifice to God. On the other hand, the entire (first) ram is burned on the altar, while the second ram has its fat burned on the altar, while the leftovers are given as food to the priests. I'll save my explanations of these rituals for now, because we will re-encounter them later.
The terms sin offering, wave offering, heave offering (NASB) and burnt offering are specific technical terms that are defined later in Leviticus, which prescribes the proper way to offer various types of sacrifices. Again, I will explain these when we reach Leviticus, because if I explain them now I'll have nothing to say then. :)
The priests eat from the sacrifice. This is the beginning of a professional ministry, because their fulltime job will be the maintenance of the tabernacle and the performing of sacrifices. It is only last chapter that an Israelite priesthood was introduced at all, and now they are to be considered a professional (and permanent, hereditary) class This is a significant shift, and it happens in parallel with the growing complexity of the LORD's religious rituals. That is, a fulltime ministry is only required because there are so many tasks that the LORD is commanding to be done. Assigning these tasks to a specific minister is efficient, yet it denies access to the holy place to the vast majority of the Israelites, as the family of Aaron will be less than 1% of the total population.
The LORD further institutes permanent, daily sacrifices. We aren't told where the animals are to come from, but I would guess either donations or some sort of national tax. Although it's not stated here, I believe the priests also eat from the daily sacrifices (and not just the sacrifices for their consecration), which will become their daily provision while they minister by maintaining the lampstand and also overseeing the sacrifices and other rituals.
We are not told what is the purpose or reason for these sacrifices, but I would speculate that they are intended as a sacrifice of worship, not atonement or consecration. We have seen a few sacrifices of worship before, such as Gen 4:3-4 and arguably also Gen 22:13. In both instances, there is no obvious sign that the men offering sacrifices are atoning for anything, they are just offering sacrifices as a token or respect or worship to God. I believe it's similar here because there is nothing specifically being atoned. I believe this is partly the basis for God's "dwell[ing] among the sons of Israel". Alternatively, one could view it as sacrificial atonement for sins the Israelites commit on a daily basis, in which case they would have daily atonement in addition to the annual Passover festival.
However, I think there's a lingering question, which is, "why does a sacrifice of worship make sense?" I already tried to address the reasoning of substitutionary atonement back in Ex 12, and I think I was able to find a sort of logic in the sense that one can transfer one's sins to the animal and then kill the animal. In this case, for the daily sacrifices, there is no apparent "transfer", so what purpose does it serve that an animal die? I honestly don't have a good answer for this, but I have some ideas. Back in Gen 4, we saw that Cain offered of the fruit that he grew from the soil, while Abel offered fat from the flocks he raised. In both cases, they offered food which was the direct result of their labor. I think this represents the duality between the results of labor and the food that we need to eat to live. These two concepts are intertwined first in Gen 3 when God says that it is by harsh labor that man shall bring plants from the ground, which he shall eat. This is analogous to the harsh labor required to shepherd a flock and eat the slaughtered lambs of that flock. This further explains why there are sacrifices of bread and oil (and in v. 40, wine) in addition to animals, because this represents the approximate diet of the Hebrews. There was no sacrifice of bread in the Passover or when the covenant is confirmed in Ex 24, because those were sacrifices of atonement, while these are sacrifices of consecration and worship.
Like the offering of firstfruits, I believe the sacrifices in this chapter are intended as an acknowledgement that God is the source of all the food we eat, that our success comes from God. God sends down rain and causes the sun to shine, and provides us all the tools we need to have a successful crop. Offering a sacrifice to God, then, shows that we recognize our dependence on him and his favor towards us.
Does it take a sacrifice to acknowledge one's dependence on God? I'd lean towards no, I don't think it does. I think the sacrifice here is instituted in part to teach the Israelite's their dependence on God, possibly more than as an acknowledgement. And why a sacrifice? Probably the logic is, "from God this came, to God shall it return."