The Lord declares that He has come, not to loosen the bonds of the Law, but to bring them to their true completion. For the Law was given that it might be fulfilled, not destroyed; and it is fulfilled when what it commands is accomplished through love. Thus, what was formerly enjoined as an outward act is now perfected by an inward disposition of the heart. The Law is not diminished, but elevated, when it is brought from the letter to the spirit.

When He says that not one jot or one tittle shall pass away until all is fulfilled, He teaches the firmness and unchangeable truth of God’s commands. For heaven and earth signify the stability of the created order, and so long as they remain, the truth of the Law remains also. Nothing that has been commanded is idle or superfluous; all tends toward that end in which it is fulfilled.

He then distinguishes between those who break the commandments and those who keep them, not by excluding the former entirely from the kingdom, but by showing their lesser place within it. For to break the least commandments and to teach others so is to belong to the kingdom in a defective manner, while to do and to teach is the mark of true greatness. Doctrine without life is empty, and life without instruction is hidden; but when both are joined, righteousness shines fully.

Finally, He declares that the righteousness of the scribes and Pharisees is insufficient, because it remains on the surface of actions and does not penetrate to the inward man. They restrain the hand, but do not purify the heart; they avoid outward transgression, but do not uproot inward desire. Therefore, a righteousness that exceeds theirs must descend into the will itself, so that the whole man may be ordered to God. From this point onward, the Lord begins to unfold how the Law is fulfilled within, by correcting not only deeds but intentions, and by drawing all obedience toward the perfection of love.

Source: St. Augustine, On the Lord’s Sermon on the Mount,  Book I, Chapters 8–9