First, I used the Greek text of the book of Hebrews (the oldest existing copies of the Apostolic Writings are written in Greek) and searched for the phrase "order of Melchizedek". The Greek word that is translated as "order" [taxin] does not mean "a monastical society" like the Benedictine or Franciscan monks. It means "a sequence" or "a succession". We get our English word taxonomy (an ordered structure) from the same source. Hebrews 5:6 (quoting Psalm 110:4) states, "You are a priest forever according to the sequence/succession of Melchizedek." So we know this priesthood has a sequence or succession associated with it.
Second, the name "Melchizedek" comes from a Hebrew phrase [mal'chi tzedek] which means "my king is righteous". The writer of the book of Hebrews plays on that phrase when he says in Hebrews 7:2 that the name means "king of righteousness". So we know from the title that this priesthood (unlike the Levitical priesthood) has something to do with righteousness.
The third and final clue was the most puzzling. I uncovered it when searching out an unrelated topic in Peter's second epistle. The Greek of 2 Peter 2:5 reads, "...and the old world did not spare, but the eighth person, Noah, of righteousness a preacher, did keep, a flood on the world of the impious having brought...". Most translators take this verse to mean that Noah was the eighth of all the people saved on the ark, but the phrasing of the Greek text does not imply that.
In this passage, Peter is making the point that G-d knows how to rescue the godly from temptation and keep the unrighteous under punishment for the day of judgment (2 Peter 2:9). He notes explicitly that G-d protected Noah, the eighth person who was a preacher of righteousness, during the Flood.
This begged the question... who were the previous seven preachers of righteousness?
Noah wasn't in the eighth generation of humankind. According to Genesis chapter 5, he was in the tenth:
(1) Adam, (2) Seth, (3) Enosh, (4) Kenan, (5) Mahalalel, (6) Jared, (7) Enoch, (8) Methuselah, (9) Lamech, (10) Noah.
At first, I thought this was a dead end in my search, but I noticed one other detail when I charted out the lifespans of those ten generations:
I: Adam (1) lived for 930 years from Creation until his death.
II: When Adam died, Seth (2) was next in succession.
III: When Seth died, Enosh (3) took over.
IV: When Enosh died, Kenan (4) took his place.
V: When Kenan died, Mahalalel (5) filled his role.
VI: When Mahalalel died, Jared (6) took that role.
VII: When Jared died, Methuselah (8) took his place because Enoch (7) had been "taken" (Genesis 5:24) and was thus skipped in the line of succession.
VIII: When Methuselah died, Noah (10) took the role because Lamech (9), who should have been next, had already died.
Using the details given in Scripture, we find that Noah would have been eighth in a sequential line if we began with Adam. I have this chart in my original article regarding the Mystery of Melchizedek. Although no other scenarios make Noah the eighth without Adam being the first, that also aligns with Scripture.
You also asked about Abram's tithe in Genesis chapter 14 and whether or not Abram would have had to have been the priest himself. I created a second chart (also in the Mystery of Melchizedek article) that begins with Noah and continues the same pattern of bestowing the priesthood upon each successive generation. When Abram gave a tithe to Melchizedek, Shem (Noah's son) was still alive (!) and would have held that title. The oral history of the Jewish people corroborates this idea by explicitly naming Shem as the person with the title Melchizedek in Genesis 14. Shem would have held that title until he died, and then Eber, his great-grandson, took over.
One additional note: if we trace the order of Melchizedek down through the history of the Jewish people, we find that Solomon would have also held that title for a time, and he frequently refers to himself as "the preacher" throughout the book of Ecclesiastes. Was he the "preacher of righteousness" in his generation? It is very likely.
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