The following is from pages 145-152 of Teaching Truths by Signs and Ceremonies or The Church, Its Rites and Services Explained for the People by Rev. Jas. L. Meagher (1882, New York: Russel Brothers). Previous exerpts can be found here:

Signs and Ceremonies
The Real Presence
The Incarnation
The Virgin Mary
The Redemption

"The Mass is the chief act of the Christian religion, the sacrifice offered at the altar, in which the institution of our Lord, and according to the rites of our holy religion the Body and Blood of Christ are offered as an unbloody sacrifice to the Father."

"But what is a sacrifice? Let us understand well what is the Mass. It is a sacrifice, and a sacrifice is an offering made by man to the Supreme Being, telling by that offering that we come entirely from Him our Creator, and therefore that we should be wholly consumed in the honor and the worship of the Almighty. But because it is not allowed man to sacrifice himself and take his life and offer it as a sacrifice, he takes another being, dear to him, and offers it to the Lord in place of himself. That is a sacrifice [Card. De Lugo, De Euchar, Disput. XIX, S. 1]. It is then the offering of a sensible thing, by a chosen minister, so that by destroying it he testifies to God’s supreme authority over us and our dependent on him; it must be the offering of a sensible thing; that is something which appears to our senses, whether living or inanimate, and not internal prayers; it must be offered by one with authority to sacrifice, separated from the people like Aaron and his family; it must be offered only to God, for sacrifice offered to a creature is idolatry; it must be destroyed so as to no more be useful to man; it must be offered with the intention of showing our subjection to God and his supreme power over us as our Creator. Then it is a true sacrifice."

"There must be then five things in a sacrifice in order that it may be acceptable to God. There must be the thing apt to be sacrificed; when living it is called a host, from the pagan custom of sacrificing those of the enemy taken in battle called hosts; there must be the act by which the thing is destroyed, called its immolation, from ancient times when they sprinkled meal or flour on the victim for the sacrifice; there must e the legitimate end of the sacrifice offered to the Supreme Being, for he alone is our Creator; there must be the legitimate man for that purpose, like Aaron chosen by the Lord, like Christ called by his Father, like the priests of the new Law descending by ordination from the Apostles; there must be the work of God in instituting the sacrifice, and ordaining what he will receive, then only will it be accepted by him as a witness of his supreme power over us. Hence in every sacrifice there is an altar, a priest, a victim and an immolation."

"Sacrifice is divided into three classes, according to the three ages of the world; the sacrifices of the Patriarchs from Adam to Moses, when the first born by birthright, was a priest of the Most High, and offered victims and oblations for the family; the sacrifices of the Jewish law from Moses to Christ, ordained by God where the priests descended from Aaron by birth, and the victims, the place and the ceremonies were pointed out by command of God; the sacrifice of the new law from Christ to the end of the world, established by our Lord himself, where the priests descend from Christ through the Apostles by ordination, and the Victim is himself, first offered at the last supper, his Body and Blood shed on Calvary’s cross for our redemption and for that of the whole human race, and the sacrifices of the Patriarchs and of the law of Moses were but figures and shadows ordained by God to prepare the world for the sacrifice of Calvary and of the Mass."

"The sacrifices of the old Law were of three kinds, victims, immolations and libations; the victims or hosts were sacrifices in which animals chosen by God were offered as cows, goats, calves, pigeons, swallows and turtle-doves; they were sacrificed by being killed and sometimes burned. Immolations were sacrifices of things without life, as first fruits of the earth, bread, salt and incense; they were burned, but to pieces, or changed so as to be useless for man. Libations were sacrifices of liquor, as water, oil, etc., and they were poured out, or destroyed before the Lord. These sacrifices were again of three kinds; the holocaust, thus called because the whole was consumed with fire; the sacrifice for sin, thus called because it was offered for sin, being partly burned and the rest eaten by the priests; the peace offerings, thus called for they were given to thank God for passed benefits and to ask peace in the future, a part was burned in the Tabernacle, a part was eaten by the priests and the third portion fell to the use of the giver."

"Such were the sacrifices of the old Law. In the new, there is one sacrifice, foreshown and prefigured by all these sacrifices, rites, ceremonies, libations and offerings of the old Law, and that is the sacrifice of Christ on Calvary and on our Altar. Sacrificed on Calvary, it is called the bloody sacrifice, sacrificed on our altar it is called the unbloody sacrifice; offered on the cross it is the sacrifice of redemption, for there he redeemed us; offered on our altars it is the sacrifice of application, for there the merits of Christ are applied to our souls. But there is but one Victim, one sacrifice. There is one and the same Host now offered by the ministry of the priest, the same as the one then offered on the cross, only there is a different way of sacrifice [Council of Trent, Ses. XXII. c.11]."

"Jesus Christ offered a real sacrifice at the last supper, when he instituted the blessed Eucharist and said [Luke 22:19], ‘Do this in commemoration of me.’ From what we have said; two things are necessary for the Eucharistic sacrifice, the real presence of Christ and the remembrance of his death. The real presence of Christ in the Eucharist was proved in Chapter V. The remembrance of his death is shown in the many rites and ceremonies, and the separation of the Body and Blood; for where the Blood is taken away from the body no one can live, but here there is no real death although the Body is on the altar and Blood is in the chalice, for Christ is now glorified and can die no more, but his death is figured in a mystic manner by the separation of the Body and the Blood."

"Christ instituted the last supper after eating the pascal lamb so that to the eating of the pascal lamb might succeed the eating of his body. And as the pascal lamb was a true sacrifice, so also the last supper, must be a true sacrifice. At that time Christ made a new alliance, a New Testament, like the Old, and like the Old it should be sanctioned by a sacrifice and by blood. For Moses sprinkling the blood of the calves on the people, said [Exodus 24:8]: ‘this is the blood of the covenant which the Lord hath made with you.’ In the same way the Lord alludes to the words and uses nearly the same words in establishing the new Covenant his New Testament saying [Matthew 26:28], ‘this is my blood of the New Testament.’ As Moses spoke of blood offered in sacrifice, therefore Christ also speaks of blood offered in sacrifice, of himself at the last supper."

"When at Antioch, the Apostles and Disciples gathered; ‘And as they were ministering to the Lord and fasting’ [Acts 17:2] the Holy Ghost spoke to them. Ministering in Greek signifies offering sacrifice, according to the most learned writers [Erasmus Edit. Lyons, T. vi, Cor. a Lapid], telling us that sacrifice was offered by the Apostles."

"St. Paul to turn the early Christians from Idolatry says [1 Cor. 10:18,21] ‘are not they that eat of the sacrifices partakers of the altar?…But the things which the heathens sacrifice, they sacrifice to devils and not to God. And I would not that you should be made partakers with devils. You cannot drink the chalice of the Lord and the chalice of devils. You cannot be partakers of the table of the Lord and the table of devils.’ The Apostle here compares the sacrifices offered to God and those offered to devils, the chalice of the Lord and that of the devils, the altar of God and that of idols. The early Christians then had sacrifices. Again he says [Hebrews 13:10] ‘We have an altar where of they that have no power to eat who serve the Tabernacle.’ The early Christians then had an altar different from that of the temple, but where there is an altar there is sacrifice, for it is an altar because sacrifice is offered on it."

"The Old Testament only confirms what we find in the New. It speaks of the sacrifice of bread and wine offered by Melchisadech kind of Salem, priest of the Most High, and who all the Fathers agree in saying was a figure of the priests of the New Testament, of our holy church, offering the sacrifice of the Son of God in the species of Bread and wine. We know the celebrated prophesy of Malachy against the Jews [Malachi 1:10-11]: ‘I have no pleasure in you saith the Lord of Hosts, and I will not receive a gift of your hands. For from the rising of the sun even to the going down, my name is great among the gentiles, and in every place there is sacrifice, and there is offered to my name a clean oblation, for my name is great among the gentiles saith the Lord of Hosts.’ Such are the words of the inspired writer, telling when the sacrifices of the temple would cease and the knowledge of God would spread among the gentiles, and in every placed the clean oblation of the Mass would be offered to the Lord. ‘In that day there shall be an altar of the Lord in the midst of the land of Egypt;’ [Isaiah 19:19] and speaking of the nations he says [Isaiah 66:21] ‘I will take them to be priests and Levites saith the Lord.’ This could not be meant of the altar of the temple, for the Jews had no altar till they came into the desert out of the land of Egypt, long before the days of the prophet; this could not relate to the altar or sacrifices of the Jews, for they were commanded under the severest penalties to sacrifice only in the holy place, in the tabernacle and in the temple appointed by the Lord; this taking of priests could not signify the priests of the Old Law, for they were of the tribe of Levi and the house of Aaron; it tells of the sacrifice of the priesthood, of the offering of the unbloody sacrifice among the gentile nations converted by the preaching of the gospel. No one will say that it is the sacrifice offered to the false gods of the gentiles, for the Lord calls it ‘the clean sacrifice,’ and who would think for a moment that the Lord himself would choose the priests of the idol worshipers. Neither could it be a spiritual sacrifice as prayers and pious aspirations or patient sufferings for these were from the beginning of the world, and God always received them. What then is meant by these prophecies but the sacrifice of the Mass, offered to God on every altar, from the rising to the setting of the sun, from one end to the other of the world? In every city, and hamlet, and village, and valley rise little temples where God loves to dwell, this is the sacrifice, the clean oblation seen by prophetic eye through the mists of ages before the time of Christ."

"Such is the voice of all monuments of antiquity, of all rituals of ancient churches, of all books of prayers, of those nations and those people who separated from the church from today up to the time of the Apostles, with one voice they cry out of the unbloody sacrifice of the Holy Eucharist. The Fathers of the church, these great minds who gathered up the traditions of early days, all speak of the sacrifice of the Mass; they speak of it as a sacrifice, the Host, the oblation, the Victim; they use the words, to offer, to sacrifice; they speak of priests and of altars; they compare it with that of Melchisadech, of the pascal lamb and say it is the sacrifice spoken of by the prophet Malachy; they prove that the Mass is the fulfillment of all these figures, rites and ceremonies, grand and majestic of the Jewish tabernacle and of Solomon’s temple, that all these figures are fulfilled in the Mass."

"St. Justin says [In Dialog. Cum. Tryon. Judeo]: ‘In every place sacrifice is offered; the bread of the Eucharist and the cup of the Eucharist as Malachy foretold.’"

"St. Ireneus, writes [Adv. Haer, 14 c.32]: ‘He who among creatures is bread, took it…saying, ‘This is my Body’ and the chalice in the same way His Blood.’"

"St. Augustine speaking of his dead mother says [Confes. L.ix., c.13]: ‘She desired to be remembered at the altar, no day passing without it, where the holy Victim is dispensed, because the chirograph which was against us was taken away.’"

"But why multiply authorities? All writers of every age speaking upon religious subjects, tell of the sacrifice of the Mass, tell us of the Faith that never changed. For if it had changed, would we not know the time and year? Could such a new doctrine begin without a revolution? Could any power on earth force people to believe such a thing if not taught by Christ and the Apostles, considering how people hold to old traditions and customs?"

"Thus Christ satisfied by the sacrifice of the Cross in paying the price, but that price is applied to our souls by the sacrifice of the Mass. The sacrifice of the Cross was to redeem mankind, the sacrifice of the Mass is to honor God, and to testify his supreme authority over us. The consecration and communion are the essential parts of the sacrifice, that is, without them there could be no sacrifice; the other parts, as prayers, ceremonies, kissings, movements, bows, replies, etc., were added by the Apostles and their successors make it complete, and to add to the beauty and dignity of the service."

"Compared to the sacrifice of the Cross, it does not differ, for one and the same Victim was offered in both. In this divine sacrifice of the Mass, the same Christ is offered in an unbloody manner, who offered himself in a bloody manner on the Cross; ‘One and the same Host now offered by the ministry of the priest, who offered himself on the Cross, only there is a different way of sacrificing.’ For Christ offered himself on the Cross, in the Mass he is offered by the priest; then he was mortal, capable of suffering, a bloody offering, here he is immortal, incapable of suffering, an unbloody offering; the sacrifice of Calvary did not commemorate any other, the Mass is a remembrance of the Cross; on Calvary the sacrifice was to pay the price of salvation, on the altar it is applied to our souls; that was offered once only, the Mass every day, and in every place to the end of the world. Thus they differ with regard to place, circumstance and manner [Schouppe, Theo., Dogmat., De Euchar, 302]."

"Thus the sacrifice of the Cross and of the Mass are the same in some respects and not the same in others. The Cross was the fulfillment of all these bloody sacrifices of the law of Moses; the Mass is the fulfillment of all these ceremonies of the tabernacle and of the temple. The first Mass said was that offered by our Lord himself at the last supper. He sent his disciples to prepare the place of that, the first of the sacrifices of the New Testament; they prepared that upper chamber, large and beautiful as tradition tells us, where having fulfilled the ceremonies of the law of Moses, for he came to fulfill and to complete these signs and ceremonies of the law given by God on Sinai’s top, where, with all the ceremonies of eating the pascal lamb, that figure of himself, he changed the bread and wine into His Body and His Blood. At that moment the Old Testament was of the passed and the New began; the law of the great lawgiver of the Israelites had passed away, and the law of the great lawgiver of the Christians began. The sacrifices of the temple were received no more by God, the sacrifice of the Mass was to take their place. The Old Testament was gone, the New Testament was there. The religion of truth, confined to the Jewish nation was given to the Gentiles, Christ was to die on the morrow and before his death he prepared the Mass as an everlasting remembrance of himself, that all nations and all people might keep him before their eyes till the day of doom, till the Angel’s trumpet calls the dead to judgment."

"We are to see then the Mass; we are to see the meaning of all these signs and ceremonies; we are to see the truths hidden in these things, added by the apostles and their successors to add to the dignity of the sacrifice, to excited devotion in the people, to raise their hearts to heavenly things, and to keep before the eyes of all generations, the life, sufferings and death of the Son of God."