We must recall that St. Cyprian also said the following:
"The Lord says to Peter: ‘I say to you,’ he says, ‘that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it. And to you I will give the keys of the kingdom of heaven; and whatever things you bind on earth shall be bound also in heaven, and whatever you loose on earth, they shall be loosed also in heaven’ [Matt. 16:18-19]). ... On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. Indeed, the others were also what Peter was [i.e., apostles], but a primacy is given to Peter, whereby it is made clear that there is but one Church and one chair. So too, all [the apostles] are shepherds, and the flock is shown to be one, fed by all the apostles in single-minded accord. If someone does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church?" (The Unity of the Catholic Church 4; 1st edition [A.D. 251]).
And Pope Julius I the following:
"[The] judgment [concerning Athanasius] ought to have been made, not as it was, but according to the ecclesiastical canon. It behooved all of you to write us so that the justice of it might be seen as emanating from all. ... Are you ignorant that the custom has been to write first to us and then for a just decision to be passed from this place [Rome]? If, then, any such suspicion rested upon the bishop there [Athanasius of Alexandria], notice of it ought to have been written to the church here. But now, after having done as they pleased, they want to obtain our concurrence, although we never condemned him. Not thus are the constitutions of Paul, not thus the traditions of the Fathers. This is another form of procedure, and a novel practice. ... What I write about this is for the common good. For what we have heard from the blessed apostle Peter, these things I signify to you" (Letter on Behalf of Athanasius [A.D. 341], in Athanasius, Apology Against the Arians 20-35).
Or the words of Optatus of Milevus:
"In the city of Rome the episcopal chair was given first to Peter; the chair in which Peter sat, the same who was head—that is why he is also called Cephas [‘Rock’]—of all the apostles, the one chair in which unity is maintained by all. Neither do the apostles proceed individually on their own, and anyone who would [presume to] set up another chair in opposition to that single chair would, by that very fact, be a schismatic and a sinner. . . . Recall, then, the origins of your chair, those of you who wish to claim for yourselves the title of holy Church" (The Schism of the Donatists 2:2 [A.D. 367]).
Pope Damasus I proclaimed:
"Likewise it is decreed . . . that it ought to be announced that . . . the holy Roman Church has been placed at the forefront not by the conciliar decisions of other churches, but has received the primacy by the evangelic voice of our Lord and Savior, who says: ‘You are Peter, and upon this rock I will build my Church, and the gates of hell will not prevail against it; and I will give to you the keys of the kingdom of heaven, and whatever you shall have bound on earth will be bound in heaven, and whatever you shall have loosed on earth shall be loosed in heaven’ [Matt. 16:18-19]. The first see, therefore, is that of Peter the apostle, that of the Roman Church, which has neither stain nor blemish nor anything like it" (Decree of Damasus 3 [A.D. 382]).
St. Jerome made these bold statements regarding the Apostolic See:
"I follow no leader but Christ and join in communion with none but your blessedness [Pope Damasus I], that is, with the chair of Peter. I know that this is the rock on which the Church has been built. Whoever eats the Lamb outside this house is profane. Anyone who is not in the ark of Noah will perish when the flood prevails" (Letters 15:2 [A.D. 396]).
"The church here is split into three parts, each eager to seize me for its own. . . . Meanwhile I keep crying, ‘He that is joined to the chair of Peter is accepted by me!’ . . . Therefore, I implore your blessedness [Pope Damasus I] . . . tell me by letter with whom it is that I should communicate in Syria" (ibid., 16:2).
Pope Innocent I claimed:
"If cases of greater importance are to be heard [at a council], they are, as the synod decrees and as happy custom requires, after episcopal judgment, to be referred to the Apostolic See" (Letters 2:3:6 [A.D. 408]).
"In seeking the things of God . . . following the examples of ancient tradition . . . you have strengthened . . . the vigor of your religion with true reason, for you have acknowledged that judgment is to be referred to us, and have shown that you know what is owed to the Apostolic See, if all of us placed in this position are to desire to follow the apostle himself [Peter] from whom the episcopate itself and the total authority of this name have emerged. Following him, we know how to condemn evils just as well as we know how to approve what is laudable. Or rather, guarding with your priestly office what the Fathers instituted, you did not regard what they had decided, not by human but by divine judgments, as something to be trampled on. They did not regard anything as finished, even though it was the concern of distant and remote provinces, until it had come to the notice of this See [Rome], so that what was a just pronouncement might be confirmed by the authority of this See, and thence other churches—just as all waters proceed from their own natal source and, through the various regions of the whole world, remain pure liquids of an incorrupted head. . . ." (ibid., 29:1).
Pope St. Leo proclaimed the following:
"Our Lord Jesus Christ . . . established the worship belonging to the divine religion. . . . But the Lord desired that the sacrament of this gift should pertain to all the apostles in such a way that it might be found principally in the most blessed Peter, the highest of all the apostles. And he wanted his gifts to flow into the entire body from Peter himself, as if from the head, in such a way that anyone who had dared to separate himself from the solidarity of Peter would realize that he was himself no longer a sharer in the divine mystery. . . . [You, my brothers], must realize with us, of course, that the Apostolic See—out of reverence for it, I mean—has on countless occasions been reported to in consultation by bishops even of your own province [Vienne]. And through the appeal of various cases to this see, decisions already made have been either revoked or confirmed, as dictated by long-standing custom" (Letters 10:2-3 [A.D. 445]).
Pope St. Gregory, mentioned above, does seem to interpret the "Feed my sheep" in the later Roman fashion:
"Your most sweet holiness, [Bishop Eulogius of Alexandria], has spoken much in your letter to me about the chair of Saint Peter, prince of the apostles, saying that he himself now sits on it in the persons of his successors. And indeed I acknowledge myself to be unworthy . . . I gladly accepted all that has been said, in that he has spoken to me about Peter’s chair, who occupies Peter’s chair. And, though special honor to myself in no wise delights me . . . who can be ignorant that holy Church has been made firm in the solidity of the prince of the apostles, who derived his name from the firmness of his mind, so as to be called Peter from petra. And to him it is said by the voice of the Truth, ‘To you I will give the keys of the kingdom of heaven’ [Matt. 16:19]. And again it is said to him, ‘And when you are converted, strengthen your brethren’ [Luke 22:32]. And once more, ‘Simon, son of John, do you love me? Feed my sheep’ [John 21:17]" (Letters 40 [A.D. 597]).
(Above quotes from http://www.catholic.com/library/Authority_of_the_Pope_Part_1.asp
Again from Optatus:
"For the good of unity Blessed Peter deserved to be preferred before the rest, and alone received the keys of the Kingdom of Heaven, that he might communicate them to the rest." Optatus of Mileve, The Schism of Donatists, 7:3 (c.A.D. 367).
Again St. Jerome, writing to the Apostolic See:
"GÃƒÆ’Ã¢â‚¬Â¡Ãƒâ€šÃ‚ÂªI think it my duty to consult the chair of Peter, and to turn to a church whose faith has been praised by PaulGÃƒÆ’Ã¢â‚¬Â¡Ãƒâ€šÃ‚ÂªThe fruitful soil of Rome, when it receives the pure seed of the Lord, bears fruit an hundredfoldGÃƒÆ’Ã¢â‚¬Â¡Ãƒâ€šÃ‚ÂªMy words are spoken to the successor of the fisherman, to the disciple of the cross. As I follow no leader save Christ, so I communicate with none but your blessedness, that is with the chair of Peter. For this, I know, is the rock on which the church is built! This is the house where alone the paschal lamb can be rightly eaten. This is the ark of Noah, and he who is not found in it shall perish when the flood prevails.” Jerome, To Pope Damasus, Epistle 15:1-2 (A.D. 375).
"Although the tradition of the Fathers has attributed to the Apostolic See so great authority that none would dare to contest its judgments...For (Peter) himself has care over all the Churches, and above all that in which he sat nor does he suffer any of its privileges or decisions to be shaken" Pope Zosimus [regn A.D. 417-418 ],To Aurelius and the Council of Carthage, Epistle 12 (A.D. 418).
"For it has never been allowed to discuss again what has once been decided by the Apostolic See." Pope Boniface [regn A.D. 418-422], To Rufus Bishop of Thessalonica, Epistle 13 (A.D. 422).
(The above quotes taken from http://scripturecatholic.com/primacy_of_peter.html#tradition_I...many
Sergius, Metropolitain of Cyprus, while writing to Pope Theodore, proclaims the following:
O Holy Head, Christ our God hath destined thy Apostolic See to be an immovable foundation and a pillar of the Faith. For thou art, as the Divine Word truly saith, Peter, and on thee as a foundation-stone have the pillars of the Church been fixed. (Sergius Ep. ad Theod. lecta in Sess. ii. Concil. Lat. anno 649)
And as for my final quote, I look to St. Maximus the Confessor (an Eastern Father):
The extremities of the earth, and everyone in every part of it who purely and rightly confess the Lord, look directly towards the Most Holy Roman Church and her confession and faith, as to a sun of unfailing light awaiting from her the brilliant radiance of the sacred dogmas of our Fathers, according to that which the inspired and holy Councils have stainlessly and piously decreed. For, from the descent of the Incarnate Word amongst us, all the churches in every part of the world have held the greatest Church alone to be their base and foundation, seeing that, according to the promise of Christ Our Savior, the gates of hell will never prevail against her, that she has the keys of the orthodox confession and right faith in Him, that she opens the true and exclusive religion to such men as approach with piety, and she shuts up and locks every heretical mouth which speaks against the Most High. (Maximus, Opuscula theologica et polemica, Migne, Patr. Graec. vol. 90)
(The above two are taken from http://web.globalserve.net/~bumblebee/ecclesia/patriarchs.htm#Cyprus
, which includes many more similar quotes).