First, there’s this note to the text…
The Latin text of this difficult but important clause is, “Ad hanc enim ecclesiam propter potiorem principalitatem necesse est omnem convenire ecclesiam.” Both the text and meaning have here given rise to much discussion. It is impossible to say with certainty of what words in the Greek original “potiorem principalitatem” may be the translation. We are far from sure that the rendering given above is correct, but we have been unable to think of anything better. [A most extraordinary confession. It would be hard to find a worse; but take the following from a candid Roman Catholic, which is better and more literal: “For to this Church, on account of more potent principality, it is necessary that every Church (that is, those who are on every side faithful) resort; in which Church ever, by those who are on every side, has been preserved that tradition which is from the apostles.” (Berington and Kirk, vol. i. p. 252.) Here it is obvious that the faith was kept at Rome, by those who resort there from all quarters. She was a mirror of the Catholic World, owing here orthodoxy to them; not the Sun, dispensing her own light to others, but the glass bringing their rays into a focus. See note at end of book iii.] A discussion of the subject may be seen in chap. xii. of Dr. Wordsworth’s St. Hippolytus and the Church of Rome.
ccel.org/fathers2/ANF-01/footnote/fn73.htm#P7320_1946616
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Now, certainly, he continues and speak of all the ancient churches built by apostles hold also the correct doctrines, however, it seems, that Rome for certain reasons is the one that every church ought to agree with, if there are some dispute concerning a certain teaching.
You missed the opening to this passage
“2. Since, however, it would be very tedious, in such a volume as this, to reckon up the successions of all the Churches, we do put to confusion all those who, in whatever manner, whether by an evil self-pleasing, by vainglory, or by blindness and perverse opinion, assemble in unauthorized meetings; [we do this, I say, ] by indicating that tradition derived from the apostles, of the very great, the very ancient, and universally known Church founded and organized at Rome by the two most glorious apostles, Peter and Paul; as also [by pointing out] the faith preached to men, which comes down to our time by means of the successions of the bishops.”
(Ibid.)
Note that he says because of Peter and Paul it is ‘pre-eminenent’ and it is an ‘example’ of the other churches. He is in effect saying “I could list all the churches which hold the faith handed down by the Apostles, however to save time, this church, founded by Peter AND Paul will serve as the example…”
He doesn’t accord Rome any superiority in power, and in fact is stating that the example of Rome must apply equally to all the other churches that have likewise kept to Christian tradition. They are thus all equal. Rome is an archetype. An ‘example’. Not the ‘leader’. Because he immediately brings it back to being the faith held by the churches everywhere, founded by all the Apostles.
The RC Church version would be that “Rome is pre-eminent because it was founded by Peter”
His version is “Rome is pre-eminent because if was founded by Peter and Paul” That is, it’s position is not dependent upon Peter alone, but because it’s the only church that was associated with two great founders. And it’s still an ‘example’ for describing all the other churches founded by all the other Apostles.
“Matthew also issued a written Gospel among the Hebrews in their own dialect, while Peter **and ** Paul were preaching at Rome, and laying the foundations of the Church.”
Irenaus, “Against Heresies”, Book III.1.1 (quoted at
ccel.org/fathers2/ANF-01/anf01-60.htm#P7297_1937859)
And this is one of the sources that says Linus NOT Peter was the first bishop of Rome.
The blessed apostles, then, having founded and built up the Church, committed into the hands of Linus the office of the episcopate."
Ibid. Book III.3.2-3
That is, Linus gets his office from the Apostles (plural). Paul and Peter are equal.
Eusebius also accepts this role for Linus "The blessed apostles having founded and established the church, entrusted the office of the episcopate to Linus. Paul speaks of this Linus in his Epistles to Timothy.
Eusebius “The History of the Church” Book V Chapter VI. Catalogue of the Bishops of Rome cited at
ccel.org/fathers2/NPNF2-01/Npnf2-01-10.htm#P2930_1381485
SO I agree, as do all Orthodox that Rome had a kind of Primacy. It is not because of Peter ALONE (which is the Roman Catholic argument). We have one section of verse we don’t know the original Greek.
In short Irenaus is saying that the Apostles went about building churches. Peter AND Paul did so in Rome. They are great men. They together founded the church there. They appointed Linus. Because they are associated with Rome he will use Rome as the example of his argument for arguing that the faith was held by all the churches founded by the Apostles.