One of the proofs of Calvin's Calvinism is the fact that the death of Christ is tied to the impetration of the death of Christ, and Calvin recognizes that the impetration of the death of Christ was only for the elect. Therefore, the proof concludes, Calvin also implicitly recognized that the death of Christ was only for the elect.
In order to understand the argument, one must understand the sacrificial system. After all, the atonement is principally a sacrifice to satisfy divine Justice and reconcile us to God. In the sacrificial system, the priest kills an animal and presents it to God. The priest presents it to God on behalf of an individual or group. We call the animal the victim and the person on whose behalf the sacrifice is offered the beneficiary. The killing of the animal is the death of the victim, its presentation to God is the impetration of the death of the victim.
In Christ's sacrifice, Christ is both the priest and victim. We, the elect, are the beneficiaries. Now, Christ is a sufficient sacrifice to be offered for as large a group of beneficiaries as Christ wishes from one person to more people than there are atoms in the universe. But when we say that Christ is "offered for us" we are saying that Christ impetrated his death on our behalf, rendering God propitious toward us. In that sense, Christ is the propitiation for our sins.
At least one of the neo-Amyraldians has acknowledged that if Calvin links (as inseparable) the death of Christ and the impetration of Christ's death, then this proof suffices to show that Calvin held limited atonement. However, this man (Steve Costley of "Controversial Calvinism") has argued that Calvin does not so link Christ's death and impetration (link to his argument).
Mr. Costley makes his argument from a passage of Calvin that is not actually germane to the issue of the relation of Christ's death and its impetration. When we examine the places where Calvin considers the issue, we do see that he links them in an inseparable way.
There are at least a few places where this can be seen in Calvin. One way that it can be seen is in Calvin's discussion of assurance:
This so great an assurance; which dares to triumph over the devil, death, sin, and the gates of hell, ought to lodge deep in the hearts of all the godly; for our faith is nothing, except we feel assured that Christ is ours, and that the Father is in him propitious to us.- Calvin, Commentary on Romans 8:34 (link)
As you can see from this example, Calvin places assurance in the linking of us being united with Christ and the Father being "in him propitious to us." The Amyraldian and Arminian views essentially allege that God is already propitious toward all mankind. If Calvin held such a view, then knowledge of God's propitiation would not be a ground of assurance of salvation, since God is also propitious (according to the Amyraldians and Arminians) to everyone, even those in hell.
Calvin goes on to make this link even stronger in his subsequent comments in the same section:
Who intercedes, etc. It was necessary expressly to add this, lest the Divine majesty of Christ should terrify us. Though, then, from his elevated throne he holds all things in subjection under his feet, yet Paul represents him as a Mediator; whose presence it would be strange for us to dread, since he not only kindly invites us to himself, but also appears an intercessor for us before the Father. But we must not measure this intercession by our carnal judgment; for we must not suppose that he humbly supplicates the Father with bended knees and expanded hands; but as he appears continually, as one who died and rose again, and as his death and resurrection stand in the place of eternal intercession, and have the efficacy of a powerful prayer for reconciling and rendering the Father propitious to us, he is justly said to intercede for us.- Calvin, Commentary on Romans 8:34 (link)
Notice how in this discussion Calvin explains that Christ's death and resurrection "stand in the place of eternal intercession." It is hard to imagine Calvin using stronger linking short of saying that the death and impetration are inseparable. Calvin actually equates here the death of Christ and the intercession of Christ, and Calvin is clear (elsewhere, as well as here) that the intercession is specific to the elect. Calvin is essentially saying that Christ intercedes for us by dying for us.
Calvin says the same thing in his Institutes:
Yet we do not dream that he intercedes for us in suppliant prostration at the Father's feet; but we apprehend, with the apostle, that he appears in the presence of God for us in such a manner, that the virtue of his death avails as a perpetual intercession for us; yet so as that, being entered into the heavenly sanctuary, he continually, till the consummation of all things, presents to God the prayers of his people, who remain, as it were, at a distance in the court.- Calvin, Institutes of the Christian Religion, Book III, Chapter XX, Section XX (John Allen, translator)
This becomes even more clear when the remainder of that section, and particularly the quotations from Augustine are considered:
Now, the cavil of the sophists is quite frivolous, that Christ is the Mediator of redemption, but believers of intercession; as if Christ, after performing a temporary mediation, had left to his servants that which is eternal and shall never die. They who detract so diminutive a portion of honour from him, treat him, doubtless, very favourably. But the Scripture, with the simplicity of which a pious man, forsaking these impostors, ought to be contented, speaks very differently; for when John says, "If any man sin, we have an Advocate with the Father, Jesus Christ," does he only mean that he has been heretofore an Advocate for us, or does he not rather ascribe to him a perpetual intercession? What is intended by the assertion of Paul, that he "is even at the right hand of God, and also maketh intercession for us?" And when he elsewhere calls him the "one Mediator between God and man," does he not refer to prayers, which he has mentioned just before? For having first asserted that intercessions should be made for all men, he immediately adds, in confirmation of that idea, that all have one God and one Mediator. Consistent with which is the explanation of Augustine, when he thus expresses himself: "Christian men in their prayers mutually recommend each other to the Divine regard. That person, for whom no one intercedes, while he intercedes for all, is the true and only Mediator. The apostle Paul, though a principal member under the Head, yet because he was a member of the body of Christ, and knew the great and true High Priest of the Church had entered, not typically, into the recesses within the veil, the holy of holies, but truly and really into the interior recesses of heaven, into a sanctuary not emblematical, but eternal, — Paul, I say, recommends himself to the prayers of believers. Neither does he make himself a mediator between God and the people, but exhorts all the members of the body of Christ mutually to pray for one another; since the members have a mutual solicitude for each other; and if one member suffers, the rest sympathize with it. And so should the mutual prayers of all the members, who are still engaged in the labours of the present state, ascend on each other's behalf to the Head, who is gone before them into heaven, and who is the propitiation for our sins. For if Paul were a mediator, the other apostles would likewise sustain the same character; and so there would be many mediators; and Paul's argument could not be supported, when he says, 'For there is one God, and one Mediator between God and men, the man Christ Jesus; in whom we also are one, if we keep the unity of the Spirit in the bond of peace.'" Again, in another place: "But if you seek a priest, he is above the heavens, where he now intercedes for you, who died for you on earth."- Calvin, Institutes of the Christian Religion, Book III, Chapter XX, Section XX (John Allen, translator)
Again, we see the close linking (both in Augustine and Calvin) of Christ's death and intercession. Hopefully, these quotations suffice to illustrate the point, although perhaps others could be adduced if someone were inclined to be argumentative or to doubt the clarity of Calvin's comments above.
Now, while this may settle the question of historical theology as to what Calvin (and Augustine) believed about the scope and extent of the atonement, it is more important to note that Calvin is rightly dividing the word of truth here. It is more important that this doctrine of the limited atonement is Scriptural, than that it was held by Calvin or Augustine. We don't accept the doctrine because those men taught it: we hold that doctrine because Scriptures teach it. I hope that no one will make the mistake of reading this and concluding that we hold to the doctrine of limited atonement simply because Calvin (or Augustine) taught it. No, they are aids to our understanding, but Scripture alone is our rule of faith.
UPDATE: Some readers may want to compare Beveridge's translation of Calvin's Institutes at the relevant section:
Moreover, the Sophists are guilty of the merest trifling when they allege that Christ is the Mediator of redemption, but that believers are mediators of intercession; as if Christ had only performed a temporary mediation, and left an eternal and imperishable mediation to his servants. Such, forsooth, is the treatment which he receives from those who pretend only to take from him a minute portion of honour. Very different is the language of Scripture, with whose simplicity every pious man will be satisfied, without paying any regard to those importers. For when John says, "If any man sin, we have an advocate with the Father, Jesus Christ the righteous," (1 John 2:1), does he mean merely that we once had an advocate; does he not rather ascribe to him a perpetual intercession? What does Paul mean when he declares that he "is even at the right hand of God, who also maketh intercession for us"? (Rom. 8:32). But when in another passage he declares that he is the only Mediator between God and man (1 Tim. 2:5), is he not referring to the supplications which he had mentioned a little before? Having previously said that prayers were to be offered up for all men, he immediately adds, in confirmation of that statement, that there is one God, and one Mediator between God and man. Nor does Augustine give a different interpretation when he says, "Christian men mutually recommend each other in their prayers. But he for whom none intercedes, while he himself intercedes for all, is the only true Mediator. Though the Apostle Paul was under the head a principal member, yet because he was a member of the body of Christ, and knew that the most true and High Priest of the Church had entered not by figure into the inner veil to the holy of holies, but by firm and express truth into the inner sanctuary of heaven to holiness, holiness not imaginary, but eternal, he also commends himself to the prayers of the faithful. He does not make himself a mediator between God and the people, but asks that all the members of the body of Christ should pray mutually for each other, since the members are mutually sympathetic: if one member suffers, the others suffer with it. And thus the mutual prayers of all the members still laboring on the earth ascend to the Head, who has gone before into heaven, and in whom there is propitiation for our sins. For if Paul were a mediator, so would also the other apostles, and thus there would be many mediators, and Paul's statement could not stand, ëThere is one God, and one Mediator between God and men, the man Christ Jesus;' in whom we also are one if we keep the unity of the faith in the bond of peace," (August. Contra Parmenian, Lib. 2 cap. 8). Likewise in another passage Augustine says, "If thou requirest a priest, he is above the heavens, where he intercedes for those who on earth died for thee," (August. in Ps. 94) imagine not that he throws himself before his Father's knees, and suppliantly intercedes for us; but we understand with the Apostle, that he appears in the presence of God, and that the power of his death has the effect of a perpetual intercession for us; that having entered into the upper sanctuary, he alone continues to the end of the world to present the prayers of his people, who are standing far off in the outer court.- Calvin, Institutes of the Christian Religion, Book III, Chapter XX, Section XX (Henry Beveridge, translator)
Or Battles' translation:
This babbling of the Sophists is mere nonsense: that Christ is the Mediator of redemption, but believers are mediators of intercession. As if Christ had performed a mediation in time only to lay upon his servants the eternal and undying mediation! They who cut off so slight a portion of honor from him are, of course, treating him gently! Yet Scripture speaks far differently, disregarding these deceivers, and with a simplicity that ought to satisfy a godly man. For when John says, "If anyone sins, we have an advocate with the Father, Christ Jesus"[1 John 2:1], does he mean that Christ was an advocate for us once for all, or does he not rather ascribe to him a constant intercession? Why does Paul affirm that he "sits at the right hand of the Father and also intercedes for us" [Romans 8:34]? But when, in another passage, Paul calls him "the sole mediator between God and man" [1 Timothy 2:5], is he not referring to prayers, which were mentioned shortly before [1 Timothy 2:1-2]? For, after previously saying that intercession is to be made for all men, Paul, to prove this statement, soon adds that "there is one God, and... one mediator" [1 Timothy 2:5].- Calvin, Institutes of the Christian Religion, Book III, Chapter XX, Section XX (Ford Lewis Battles, translator)
Augustine similarly explains it when he says: "Christian men mutually commend one another by their prayers. However, it is he for whom no one intercedes, while he intercedes for all, who is the one true Mediator." The apostle Paul, although an eminent member under the Head, yet because he was a member of Christ's body, and knew that the greatest and truest priest of the church had not figuratively entered the inner precincts of the veil to the Holy of Holies but through express and steadfast truth had entered the inner precincts of heaven to a holiness real and eternal, also commends himself to the prayers of believers [Romans 15:30; Ephesians 6:19; Colossians 4:3]. And he does not make himself mediator between the people and God, but he asks that all members of Christ's body mutually pray for one another, "since the members are concerned for one another, and if one member suffers, the rest suffer with it" [1 Corinthians 12:25-26, Cf. Vg.]. And thus the mutual prayers for one another of all members yet laboring on earth rise to the Head, who has gone before them into heaven, in whom "is propitiation for our sins" [1 John 2:2, Vg.]. For if Paul were mediator, so also would the rest of the apostles be; and if there were many mediators, Paul's own statement would not stand, in which he had said: "One God, one mediator between God and men, the man Christ" [I Tim. 2:5], "in whom we also are one" [Romans 12:5], "if we maintain unity of faith in the bond of peace" [Ephesians 4:3]. Likewise, in another passage Augustine says: "But if you seek a priest, he is above the heavens, where he is making intercession for you, who died for you on earth." [Cf.Hebrews 7:26 ff.]
But we do not imagine that he, kneeling before God, pleads as a suppliant for us; rather, with the apostle we understand he so appears before God's presence that the power of his death avails as an everlasting intercession in our behalf [cf. Romans 8:34], yet in such a way that, having entered the heavenly sanctuary, even to the consummation of the ages [cf. Hebrews 9:24 ff.], he alone bears to God the petitions of the people, who stay far off in the outer court.