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Animal sacrifice was intended to bring the mind of the offerer to consider the life that God requires them to lead to be acceptable to him. The animals had to be without blemish.
It began in Eden when Adam and Eve disobeyed God. They had sinned and needed a covering for their sin. We might reasonably suppose that this was provided symbolically by the skin of an animal (that had previously been declared very good Genesis 1:31) to cover their nakedness.
Genesis 3:21 Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them.
The providing of a perfect sacrifice to cover mankind’s sin and bring about rconciliation is central to the purpose of God. The sacrifice to be provided was God’s only begotten son.
John Baptist bears witness:
John 1:29... John seeth Jesus coming unto him, and saith, Behold the Lamb of God, which taketh away the sin of the world.
The animals which were acceptable for sacrifice were known as ‘clean’ animals. Noah was told to take animals in twos into the ark, but to take seven of each of the ‘clean’ animals.
Genesis 7:2 Of every clean beast thou shalt take to thee by sevens, the male and his female: and of beasts that are not clean by two, the male and his female.
When God gave the Law to Moses at Sinai, the characteristics that make an animal clean and acceptable to God are spelled out.
Leviticus 11:3 Whatsoever parteth the hoof, and is clovenfooted, and cheweth the cud, among the beasts, that shall ye eat.
The characteristics of dividing the hoof and chewing the cud are easily understood to represent being sure footed, rightly dividing or understanding the word of God and meditating upon it on the part of the offerer. Only those which possessed both characteristics were acceptable.
God instructs Joshua:
Joshua 1:8 This book of the law shall not depart out of thy mouth; but thou shalt meditate therein day and night, that thou mayest observe to do according to all that is written therein: for then thou shalt make thy way prosperous, and then thou shalt have good success.
These were the characteristics of Jesus Christ.
Psalms 119:97 O how love I thy law! it is my meditation all the day. 98 Thou through thy commandments hast made me wiser than mine enemies: for they are ever with me. 99 I have more understanding than all my teachers: for thy testimonies are my meditation. 100 I understand more than the ancients, because I keep thy precepts.101 I have refrained my feet from every evil way, that I might keep thy word.
Paul tells us:
Galatians 3:24 Wherefore the law was our schoolmaster to bring us unto Christ, that we might be justified by faith. 25 But after that faith is come, we are no longer under a schoolmaster.
The blood of animals could never take away sin. The sacrifices needed faith on the part of the offerer that they were representative of the one to be provided by God. Salvation has always required faith in the promise of God to provide a saviour.
Hebrews 10:4 For it is not possible that the blood of bulls and of goats should take away sins. 5 Wherefore when he cometh into the world, he saith, Sacrifice and offering thou wouldest not, but a body hast thou prepared me: 6 In burnt offerings and sacrifices for sin thou hast had no pleasure. 7 Then said I, Lo, I come (in the volume of the book it is written of me,) to do thy will, O God. 8 Above when he said, Sacrifice and offering and burnt offerings and offering for sin thou wouldest not, neither hadst pleasure therein; which are offered by the law; 9 Then said he, Lo, I come to do thy will, O God. He taketh away the first, that he may establish the second. 10 By the which will we are sanctified through the offering of the body of Jesus Christ once for all.
Because Christ fulfilled the Law as the perfect sacrifice, the Mosaic Covenant is annulled and replaced with a New Covenant in Christ (He taketh away the first, that he may establish the second). The voluntary sacrifice of Christ who was sinless really can take away the sin of those who believe and obey the gospel.
1 Peter 1:18...ye were not redeemed with corruptible things, as silver and gold, from your vain conversation (way of life) received by tradition from your fathers; 19 But with the precious blood of Christ, as of a lamb without blemish and without spot: 20 Who verily was foreordained (Greek – ‘proginosko’ - foreknown) before the foundation of the world, but was manifest in these last times for you,
The animals were not slain instead of the offerer, as some believe, but were intended to underline the necessity for living righteously before God. Animals were sinless and could represent a sinless life. The blood of the sacrifices was always dealt with as most holy, being representative of how God views the lives of those who serve him faithfully. In the antitype, Jesus did not die instead of us, for we all still die, rather he surrendered his perfect life voluntarily to redeem us from the curse of sin and death.
John 15:13 Greater love hath no man than this, that a man lay down his life for his friends. 14 Ye are my friends, if ye do whatsoever I command you.
Those who believe the gospel and are baptised into Christ will be raised from the dead when he returns. By belief and baptism we pass from Adam into Christ.
Paul explains what is to happen:
1 Corinthians 15:21 For since by man came death, by man came also the resurrection of the dead. 22 For as in Adam all die, even so in Christ shall all be made alive. 23 But every man in his own order: Christ the firstfruits; afterward they that are Christ’s at his coming. 24 Then cometh the end, when he shall have delivered up the kingdom to God, even the Father; when he shall have put down all rule and all authority and power. 25 For he must reign, till he hath put all enemies under his feet. 26 The last enemy that shall be destroyed is death. 27 For he hath put all things under his feet. But when he saith all things are put under him, it is manifest that he is excepted, which did put all things under him. 28 And when all things shall be subdued unto him, then shall the Son also himself be subject unto him that put all things under him, that God may be all in all.
There is a lot more to the significance of the different sacrifices but I think this should be sufficient for now to answer your question.
I hope you have found this helpful.
May God bless you,