John Calvin on the Old and New Covenants

There is one covenant that God has made to reestablish fellowship with sinful humanity, revealed in first in the old covenant (Old Testament) and then in the new covenant (New Testament). There is one people of God throughout history, bound to Him and to each other by this covenant. Christ did not come to abolish the old covenant. He came to bring it to fulfillment in a permanent form, based on His redemptive work, with greater spiritual power and new ceremonies that more clearly exhibit what the old covenant ceremonies pointed to: Christ. John Calvin remarked on this continuity between the old and new covenants, and the abiding authority of the Old Testament law, in his commentary on the Gospels.

"Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them." (Matthew 5:17)

"God had, indeed, promised a new covenant at the coming of Christ; but had, at the same time, showed, that it would not be different from the first, but that, on the contrary, its design was, to give a perpetual sanction to the covenant, which he had made from the beginning, with his own people.

'I will write my law, (says he,) in their hearts,
and I will remember their iniquities no more.'
(Jeremiah 31:33, 34)

By these words he is so far from departing from the former covenant, that, on the contrary, he declares, that it will be confirmed and ratified, when it shall be succeeded by the new. This is also the meaning of Christ’s words, when he says, that he came to fulfill the law: for he actually fulfilled it, by quickening, with his Spirit, the dead letter, and then exhibiting, in reality, what had hitherto appeared only in figures. 

With respect to doctrine, we must not imagine that the coming of Christ has freed us from the authority of the law: for it is the eternal rule of a devout and holy life, and must, therefore, be as unchangeable, as the justice of God, which it embraced, is constant and uniform. With respect to ceremonies, there is some appearance of a change having taken place; but it was only the use of them that was abolished, for their meaning was more fully confirmed. The coming of Christ has taken nothing away even from ceremonies, but, on the contrary, confirms them by exhibiting the truth of shadows: for, when we see their full effect, we acknowledge that they are not vain or useless. Let us therefore learn to maintain inviolable this sacred tie between the law and the Gospel, which many improperly attempt to break. For it contributes not a little to confirm the authority of the Gospel, when we learn, that it is nothing else than a fulfillment of the law; so that both, with one consent, declare God to be their Author." (source)