Richards:

Second. Was his death, then, of vicarious import simply? or was it strictly vicarious?

That it was of vicarious import cannot reasonably be denied, if we compare it with the legal sacrifices, or attend to the express language of Scripture on the subject.

The victims under the law were vicarious offerings; they suffered in the room and stead of the offerer, and thus far there was a transfer, not of sin or guilt, strictly speaking, but of its penal effects; suffering and death, only, were transferred, and this is what is meant by putting the iniquities of the sinner upon the head of the victim, and of the victim’s bearing the iniquities of the sinner.

To suppose a literal transfer, either of sin or of punishment, would be to suppose something which is entirely unauthorized by the language of Scripture, and at the same time to involve the absurdity of making a man and even a beast guilty by proxy. Sin, guilt, ill-desert, are in the very nature of things personal; and punishment presupposes guilt, and guilt in the subject; neither the one nor the other is properly transferable. Or, to use the language of Magee: “Guilt and punishment cannot be conceived but with reference to consciousness which cannot be transferred.”

While we would maintain, therefore, that the sufferings of Christ were of vicarious import, because he suffered in the room of sinners, and bore the indications of Divine wrath for their sakes, we cannot subscribe to the opinion that they were strictly vicarious, if by this is meant that the sins of those for whom he suffered, their personal desert and their punishment were literally transferred to him. We maintain the doctrine of substitution, but not such a substitution as implies a transfer of character, and consequently of desert and punishment. This we think to be impossible; and unnecessary, if not impossible. It was enough that there should be a transfer of sufferings, and these, not exactly in kind, degree, or duration, but in all their circumstances amounting to a full equivalent in their moral effect upon the government of God. We hold that Jesus died in the room of the guilty, that though innocent himself, he was made sin for us, or treated as a sinner on our account, and in our stead; that the Lord laid on him the iniquities of us all, and that he bore our sins in his own body on the tree, by suffering what was a full equivalent to the punishment due to our offences. But this, we think, is all the substitution which the Scriptures teach, all that the nature of things will admit, and all that was necessary to effect the same moral ends in the government of God which would have been effected by inflicting on the transgressor the penal sanctions of his law.

James Richards, Lectures on Mental Philosophy and Theology (New York: Published by M.W. Dodd, 1846), 312-314.