Aquinas:

I answer that, He properly atones for an offense who offers something which the offended one loves equally, or even more than he detested the offense. But by suffering out of love and obedience, Christ gave more to God than was required to compensate for the offense of the whole human race. First of all, because of the exceeding charity from which He suffered; secondly, on account of the dignity of His life which He laid down in atonement, for it was the life of one who was God and man; thirdly, on account of the extent of the Passion, and the greatness of the grief endured, as stated above (46, 6). And therefore Christ’s Passion was not only a sufficient but a superabundant atonement for the sins of the human race; according to 1 John 2:2: “He is the propitiation for our sins: and not for ours only, but also for those of the whole world.”

Thomas Aquinas, Summa Theologica, Part 3, Q 48.2.

Luther:

1) The text, then, means to say this: The old priests washed their hands and feet externally in their lavers; but now there is to be a washing in which not hands and feet are to be washed but all sin and uncleanness is to be washed away, so that, even if someone should sin and still have many of Adam’s and Eve’s other evil inclinations in him, everything should still become clean. For it is a daily, public, free washing, that is, an eternal forgiveness of sins, which is at all times open to all sinners and unclean persons; and we say this in the Creed: “I believe in the forgiveness of sins,” and in 1 John 2:2: “Christ is the expiation for our sins, and not for ours only but also for the sins of the whole world, etc.” Martin Luther, “Minor Prophets,” in Luther’s Works, 20:332.

2) Thus truth is faith itself, which judges correctly about God, namely, that God does not look at our works and our righteousness, since we are unclean, but that He wants to be merciful to us, to look at us, to accept us, to justify us, and to save us if we believe in His Son, whom He has sent to be the expiation for the sins of the whole world (1 John 2:2). This is the true idea about God, and it is really nothing other than faith itself. By my reason I cannot understand or declare for certain that I am accepted into grace for the sake of Christ, but I hear this announced through the Gospel and take hold of it by faith. Martin Luther, “Lectures on Galatians,” in Luther’s Works, 26:238-239.¹

3) Then we shall be as we have been redeemed and purified. There must be a careful distinction between the two. Thus Peter says (1 Peter 2:24): “He Himself bore [our sins]–that we might live to righteousness.” Now that Christ has suffered in the flesh, it follows: “As He has done, so we ought to do.” Thus John says (1 John 2:9): “He who says he is in the light.” He is speaking of the imitation [of Christ]. But, on the other hand, he says (1 John 2:2): “not for our sins only but also for the sins of the whole world.” Here he is referring to the gift. Therefore you must carefully consider these two things. Heretofore we have taught in the schools that Christ is an example and a lawgiver. But about the other part, how He has been given for us, they taught nothing at all. And yet this is the most important part and the summary of what ought to be taught and known in Christ; if this is not taught, faith perishes, because righteousness is not based on the teaching of the first part. Martin Luther, “Lectures on Titus,” in Luther’s Works, 29:67-68.²

4) 2. And He is the expiation for our sins.

He does not sit at the right hand of the Father to terrify us, but He is the expiation. Nevertheless, we seek other advocates, others to render satisfaction and make expiation for our sins. Our sins are too great. They cannot be atoned for with our works; this can be done only with Christ’s bitter suffering and with the shedding of His precious blood. Sin causes heartache and depicts Christ for us differently from what He is; it shows Him to us through a colored glass. Even some teachers have done this, even the very saintly martyr Cyprian. But these things must be proclaimed to those who have been terrified, not to those who are presumptuous. Christ, who does not spurn a contrite and humble heart, wants to be the Lord and Author of life, not of sin.

But also for the sins of the whole world.

It is certain that you are a part of the world. Do not let your heart deceive you by saying: “The Lord died for Peter and Paul; He rendered satisfaction for them, not for me.” Therefore let everyone who has sin be summoned here, for He was made the expiation for the sins of the whole world and bore the sins of the whole world. For all the godless have been put together and called, but they refuse to accept. Hence it is stated in Is. 49:4: “I have labored in vain.” Christ is so merciful and kind that if it were possible, He would weep for every sinner who is troubled. Of all men He is the mildest, of all the gentlest. With every member He feels more pity than Peter felt under the rod and the blows. Take any man who is extraordinarily kind and gentle. Then you would know that Christ is much kinder to you. For just as He was on earth, so He is in heaven. Thus Christ has been appointed as the Bishop and Savior of our souls (cf. 1 Peter 2:25). But at His own time He will come as Judge. Since we see this, let us give no occasion to gratify lust. Martin Luther, “The Catholic Epistles,” in Luther’s Works, 30:236-237.

5) Besides, some of them are now beginning to preach shamelessly the blasphemous doctrine that Christ has made satisfaction only for original sin and sins prior to baptism; for sins that follow baptism we must ourselves make satisfaction. This is simply to make Turks and heathen out of Christians. It does not take into account that John in the first chapter of his First Epistle clearly says about all Christians and about himself: “But if we walk in the light, … the blood of Christ his Son cleanses us from all sin” [I John 1:7]. And in the second chapter of the First Epistle of John: “But if any one does sin, we have an advocate with the Father, Jesus Christ the righteous; and he is the expiation for our sins, and not for ours only, but also for the sins of the whole world” [I John 2:1–2]. The Epistle to the Hebrews gloriously portrays Christ’s eternal priesthood, namely, how he intercedes for us before God [Hebrews 4:14], and Paul says to the Romans in the eighth chapter: Christ “intercedes for us” before the Father [Romans 8:34]. But what could such leaders of the blind and traducers of baptism understand about these matters? Martin Luther, “The Private Mass and the Consecration of Priests,” in Luther’s Works, 38:183-184.

¹Galatians 3:7.
²Titus 2:14.

Before leaving this topic, I must briefly notice an objection to the fatherhood of God as manifested by the atonement, which may possibly be urged by some of those who hold with us limited or special destination of this great remedial provision for human guilt. The atonement, they may urge, was intended to secure the salvation, not of all the members of the human family, but exclusively of those to whom it was God s purpose savingly to apply it by the grace of His Holy Spirit. Accordingly, it furnishes, we may be told, no proof of the fatherhood of God in relation to all mankind. It shows Him to be the Father of those who are eventually saved, but of none besides them.Now it must be owned that the purposes of God, in their bearing on the atonement, as on every other subject with reference to which they may be brought into discussion, involve mysteries which are too deep for the intellect of man to fathom. But, happily, it is not necessary to enter into these mysteries in meeting the objection with which we are now called to deal. For however limited or definite may be our views of the destination of the atonement in the secret purposes of Him who orders all things after the counsel of His will, there are certain broad and patent facts respecting it, which, not the less on that account, we find ourselves obliged to believe, and in virtue of which we cannot otherwise regard it than as gloriously illustrative of the general paternity of God.

For example, we believe in its full perfection and sufficiency as an adequate propitiation for the sins of the whole world, insomuch that no other or greater atonement would have been necessary to secure the salvation of every member of our fallen race. We believe also in the unlimited freeness and universality of the offers in which it is held out to our acceptance, insomuch that there is nothing to prevent any or every sinner from seeking an interest in the blessings it has secured; or rather there is nothing to excuse any or every sinner for declining, when thus solicited, to take advantage of the offered mercy. And farther, we believe that in addressing to us these unlimited calls, the God of all grace is sincerely seeking our compliance with them, and that, in terms of His own solemn declarations, “He has no pleasure in the death of the wicked, but that the wicked should turn from his way and live,” [Ezek. 33:11]–“He will have all men to be saved, and to come unto the knowledge of the truth,” [1 Tim.2:4] and “He is not willing that any should perish, but that all should come to repentance.” [2 Peter 3:9].

Thomas Crawford, The Fatherhood of God, (Edinburgh: William Blackwood and Sons, 1868), 135-136.

Hodge:

This is what is meant when it is said, or implied in Scripture, that Christ gave Himself as a propitiation, not for our sins only, but for the sins of the whole world. He was a propitiation effectually for the sins of his people, and sufficiently for the sins of the whole world. Augustinians have no need to wrest the Scriptures. They are under no necessity of departing from their fundamental principle that it is the duty of the theologian to subordinate his theories to the Bible, and teach not what seems to him to be true or reasonable, but simply what the Bible teaches.

Charles Hodge, Systematic Theology, 2:558-9.