From “A Treatise Upon the Passion” by St. Thomas More (+1535)

In the twenty-fourth chapter of Exodus it is related that Moses, in confirmation of the old Law, put half the blood of the sacrifice into a cup, and the other half he shed upon the altar. And, after the book of the Law had been read, he sprinkled the blood upon the people and said unto them: This is the blood of the covenant that the Lord has made with you in all these words. And so was the Old Testament ratified and confirmed with blood. And in like manner was the New Testament confirmed with blood, saving that, in order to declare the greater excellence of the New Testament brought by the Son of God, above the Old Testament brought by the prophet Moses, whereas the Old Testament was ratified with the blood of a brute beast, the New Testament was ratified with the blood of a rational man, and of that man who was also God, that is to say, with the blessed blood of our holy Saviour himself. And that self-same blood did our Lord here give unto his apostles in this blessed sacrament, as he plainly declared himself, saying: This is my blood of the New Testament, or: This is the chalice of the New Testament in my blood which shall be shed for you and for many for the remission of sins. 

When our Lord said this, he declared therein the efficacy of the New Testament above the old, in that the old Law in the blood of beasts could only promise the remission of sin that was to come later. For as Saint Paul says: It was impossible that sin should be taken away by the blood of brute beasts. But the new Law with the blood of Christ does perform the thing that the old Law promised, that is, the remission of sin And therefore our Saviour said: This is the chalice of the New Testament in my blood – that is, to be confirmed in my blood – which shall be shed for the remission of sins. 

His words also declared the wonderful excellence of this new blessed sacrament above the sacrifice of the paschal lamb, in these words: For you and for many. For in these words our Saviour spoke, says Saint Chrysostom, as though he meant to say: The blood of the paschal Lamb was shed only for the first-born among the children of Israel, but this blood of mine shall be shed for the remission of the sin of all the whole world.