Abraham begat Isaac; and Isaac begat Jacob; and Jacob begat Judah and his brothers;
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Ambrose of Milan
In Luc. cap. 3. lib. iii. n. 7,8: For Abraham was the first who deserved the witness of faith; “He believed God, and it was accounted to him for righteousness.” It behoved therefore that he should be set forth as the first in the line of descent, who was the first to deserve the promise of the restoration of the Church, “In thee shall all the nations of the earth be blessed.” And it is again brought to a period in David, for that Jesus should be called his Son; hence to him is preserved the privilege, that from him should come the beginning of the Lord’s genealogy.
De Con. Evan., ii, 1: Matthew, by beginning with Christ’s genealogy, shows that he has undertaken to relate Christ’s birth according to the flesh. But Luke, as rather describing Him as a Priest for the atonement of sin, gives Christ's genealogy not in the beginning of his Gospel, but at His baptism, when John bare that testimony, “Lo, He that taketh away the sins of the world.” In the genealogy of Matthew is figured to us the taking on Him of our sins by the Lord Christ: in the genealogy of Luke, the taking away of our sins by the same; hence Matthew gives them in a descending, Luke in an ascending, series. But Matthew, describing Christ's human generation in descending order, begins his enumeration with Abraham.
Abraham begat Isaac. These two, with those who came after them, were the first patriarchs, the founders of the synagogue and people of God, and of the Kingdom of Christ. They, as types, foreshadowed Him. (See comment on Genesis , where I have unfolded their genealogies.) I will not here repeat what has been said. God constantly calls Himself the God of Abraham, Isaac, and Jacob, and even makes a boast, so to say, of this title. Wherefore He chose the posterity of Abraham, descending through Isaac and Jacob, for His own family and Church, and gave them the sign and pledge of circumcision. Wherefore God changed Abraham"s name from Abram, i.e, a high father, to Abraham, that is to say, Î°Î± ÏˆÎ± Î´Î¾Î¶Î¿, ab rab hamon, or the father of a great multitude—viz, of the believing people that should be born of him according to the flesh; in like manner as of Christ, both Jews and Gentiles, who believe in Him, are born according to the Spirit. Now Isaac—i.e. laughter—about to be offered up by hi...
But Judah is the only one mentioned by name, and that because the Lord was descended from him only. But in each of the Patriarchs we must note not their history only, but the allegorical and moral meaning to be drawn from them; allegory, in seeing whom each of the Fathers foreshewed; moral instruction in that through each one of the Fathers some virtue may be edified in us either through the signification of his name, or through his example. Abraham is in many respects a figure of Christ, and chiefly in his name, which is interpreted the Father of many nations, and Christ is Father of many believers. Abraham moreover went out from his own kindred, and abode in a strange land; in like manner Christ, leaving the Jewish nation, went by His preachers throughout the Gentiles.
Hom. iii, and Aug. City of God, 15, 15: Matthew then, desiring to preserve in memory the lineage of the Lord's humanity through the succession of His parents, begins with Abraham, saying, “Abraham begat Isaac.” Why does he not mention Ismael, his first-born? And again, “Isaac began Jacob;” why does he not speak of Esau his first-born? Because through them he could not have come down to David.
Hom. iii: Or, he names all the twelve Patriarchs that he may lower that pride which is drawn from a line of noble ancestry. For many of these were born of maidservants, and yet were Patriarchs and heads of tribes.
Isaac is interpreted, ‘laughter, 'but the laughter of the saints is not the foolish convulsion of the lips, but the rational joy of the heart, which was the mystery of Christ. For as he was granted to his parents in their extreme age to their great joy, that it might be known that he was not the child of nature, but of grace, thus Christ also in this last time came of a Jewish mother to ...
But why is it, that having mentioned Abraham, and having said that he begot Isaac, and Isaac, Jacob; and not having made any mention of his brother; when he has come to Jacob, he remembers both Judah, and his brethren? Now there are some that say, it was because of the perverseness of Esau, and of the rest that came before. But I should not say this; for if it were so, how is it that he a little after mentions such women? It being out of contraries, in this place, that His glory is manifested; not by having great forefathers, but low and of little account. For to the lofty One it is a great glory to be able to abase Himself exceedingly. Wherefore then did He not mention them? Because Saracens, and Ishmaelites, and Arabians, and as many as are sprung from those ancestors, have nothing in common with the race of the Israelites. For this cause then he passes over those in silence, and hastens on to His forefathers, and those of the Jewish people. Wherefore he says, And Jacob begot Judas a...
Matthew begins the genealogy with Abraham because he was the father of the Hebrews, and because he first received the promises that in his seed all the nations would be blessed (Gen. 22:18). Therefore it is fitting that he should give the genealogy of Christ beginning with Abraham, for the seed of Abraham is Christ by Whom all we nations have been blessed who were formerly cursed. "Abraham" means "father of the nations," and "Isaac," "joy" or "laughter." The evangelist makes no mention of the illegitimate children of Abraham, Ishmael and the others, because the Jews were not descended from them, but from Isaac.
You see that he mentioned Judah and his brothers because the twelve tribes were from them.