31 But why does St Matthew reckon in a descending, and Luke in an ascending order? I pray you give attentive ear to what the Lord may help me to say on this matter; with your minds now at ease, and disembarrassed from all the perplexity of these cavillings. Matthew descends through his generations, to signify our Lord Jesus Christ descending to bear our sins, that in the seed of Abraham all nations might be blessed. Wherefore, he does not begin with Adam, for from him is the whole race of mankind.
Nor with Noe, because from his family again, after the flood, descended the whole human race. Nor could the man Christ Jesus, as descended from Adam, from whom all men are descended, bear upon the fulfilment of prophecy; nor, again, as descended from Noe, from whom also all men are descended; but only as descended from Abraham, who at that time was chosen, that all nations should be blessed in his seed, when the earth was now full of nations. But Luke reckons in an ascending order, and does not begin to enumerate the generations from the beginning of the account of our Lord's birth, but from that place, where he relates His Baptism by John.
Now, as in the incarnation of the Lord, the sins of the human race are taken upon Him to be borne, so in the consecration of His Baptism are they taken on Him to be expiated. Accordingly, St. Matthew, as representing His descent to bear our sins, enumerates the generations in a descending order; but the other, as representing the expiation of sins, not His own, of course, but our sins, enumerates them in an ascending order. Again, St. Matthew descends through Solomon, by whose mother David sinned; St.
Luke ascends through Nathan another son of the same David, through whom he was purged from his sin. For we read, that Nathan was sent to him to reprove him, and that he might through repentance be healed. Both Evangelists meet together in David; the one in descending, the other in ascending; and from David to Abraham, or from Abraham to David, there is no difference in any one generation. And so Christ, both the Son of David and the Son of Abraham, comes up to God. For to God must we be brought back, when renewed in Baptism, from the abolition of sins.
Source: Sermons on Selected Lessons of the New Testament (New Advent)