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Cain and Abel's offerings
Abel kept sheep and offered of the firstborn of his flock and their fat portions to the LORD, while Cain worked the ground and offered of the fruit of the ground. The LORD had regard for Abel's offering but not Cain's. When Cain became angry, the LORD warned him, "If you do well, will you not be accepted? And if you do not do well, sin is crouching at the door. Its desire is contrary to you, but you must rule over it.”
(Genesis 4)
Cosmology
God accepts the offering of Abel rather than Cain because one is higher than the other in the vertical space of this cosmology (Intangible Meaning - Spiritual – Animal – Vegetable – Mineral – Meaningless Matter).
Abel was responsible for lowering the spiritual into the animal and raising the animal into the spiritual. He was a prototype of the priest who specializes in offering refined flesh to God. His name means “breath” or “bodiless meaning” for his lofty ideas without concrete facts.
Cain was responsible for lowering the vegetable into the mineral and raising the mineral into the vegetable. Thus, Cain and his offspring are prototypes of builders and smiths who specialize in making tools and vessels from refined minerals. “Cain” is connected to “nest” after his function to build a stable physical reality to host a higher spiritual identity.
In a perfect world, Cain and Abel would reach an agreement by which spiritual and material realities would have been joined correctly. Abel would cover Cain with spiritual meaning from heaven, and Cain would support Abel with physical power from the earth. Authority and power did not reach an agreement, though.
(The Language of Creation: Cosmic Symbolism in Genesis - A Commentary by Matthieu Pageau, 2018. Pages 89-92)
Noah's offering
After leaving the ark, Noah built an altar to the Lord and took some of every clean animal and every clean bird and offered burnt offerings. When the Lord smelled the pleasing aroma, He said in His heart, "I will never again curse[a] the ground because of man, for the intention of man's heart is evil from his youth. Neither will I ever again strike down every living creature as I have done. While the earth remains, seedtime and harvest, cold and heat, summer and winter, day and night, shall not cease.”
Abram's offering during the establishment of the Abrahamic covenant
When the Lord was establishing the Abrahamic covenant with Abram, Abram asked how he was to know that he is to possess the promised land. The Lord told him, “Bring me a heifer three years old, a female goat three years old, a ram three years old, a turtledove, and a young pigeon.” He brought them, cut all but the birds in half, and laid each half over against the other. When birds of prey came down on the carcasses, Abram drove them away. Abram fell into a deep sleep, and the Lord revealed the future of the nation that would descend from him.
David builds an altar at the threshing floor of Araunah the Jebusite and offers burnt offerings and peace offerings
The LORD sent a 3-day plague over Israel. David asked His hand be against him and his father's house only, and the LORD sent Gad, saying to raise an altar to the LORD on the threshing floor of Araunah the Jebusite. King David purchased the threshing floor and some oxen from Araunah for fifty shekels of silver, for he would not offer burnt offerings that cost him nothing. The LORD responded to the plea for the land and David's burnt offerings and peace offerings, and the plague in Israel was averted.