After the Babylonish exile That is, after the Jews were carried into captivity: for the Evangelist means, that the descendants of David, from being kings, then became exiles and slaves. As that captivity was a sort of destruction, it came to be wonderfully arranged by Divine providence, not only that the Jews again united in one body, but even that some vestiges of dominion remained in the family of David. For those who returned home submitted, of their own accord, to the authority of Zerubbabel. In this manner, the fragments of the royal scepter [1] lasted till the coming of Christ was at hand, agreeably to the prediction of Jacob, "The scepter shall not depart from Judah, nor a lawgiver from between his feet, until Shiloh come," (Genesis 49:10.) And even during that wretched and melancholy dispersion, the nation never ceased to be illuminated by some rays of the grace of God. The Greek word metoikesia, which the old translator renders transmigration, and Erasmus renders exile, literally signifies a change of habitation. The meaning is, that the Jews were compelled to leave their country, and to dwell as "strangers in a land that was not theirs," (Genesis 15:13.)

Footnotes

1 - "Qui avoit este mis bas, et comme rompu;" -- "which had been thrown down, and, as it were, broken."

Jechonias begat Salathiel - After Jechonias was brought to Babylon, he was put in prison by Nebuchadnezzar, where he continued till the death of this prince, and the accession of Evilmerodach, who brought him out of prison, in which he had been detained thirty-seven years, and restored him to such favor that his throne (seat) was exalted above all the kings which were with him in Babylon: Jeremiah 52:31, Jeremiah 52:32. But though he thus became a royal favorite, he was never restored to his kingdom. And, according to the prophecy of Jeremiah, Jeremiah 22:30, no man of his seed sat upon the throne of David; yet the regal line was continued through his son Salathiel, who died in Babylon: but Zorobabel, his son, returned from captivity, and by him the race of David was continued, according to Matthew, by Abiud; and, according to Luke, by Rhesa. See on Luke 3:23 (note), etc.
The term carrying away to Babylon, μετοικεσια, from μετοικεω, to change a habitation, or place of residence, would be more properly translated by the word transportation, which is here peculiarly appropriate: the change was not voluntary; they were forced away.

And after they were brought to Babylon,.... Not Jechonias, but the father of Jechonias, and the Jews.
Jechonias begat Salathiel. Not Jechonias mentioned in the former verse, but his son, called Jehoiachin, 2-Kings 24:6 and Coniah, Jeremiah 22:24 both which are rendered Jechonias by the Septuagint in 2-Chronicles 36:8 and he is so called, 1-Chronicles 3:16. Abulpharagius (c) calls him Junachir, and says he is the same who in Matthew is called Juchonia; and he asserts him to be the father of Daniel the Prophet. But here a considerable difficulty arises, how he can be said to beget Salathiel, called Shealtiel, Haggai 1:1 when he was pronounced "childless", Jeremiah 22:30. To remove which, it may be observed, that the sentence pronounced may be considered with this tacit condition or proviso, if he repented not. Now the Jews have a tradition (d) that he did repent in prison, upon which the sentence was revoked; but there is no need to suppose this, though it is not an unreasonable supposition; for the sentence does not imply that he should have no children, but rather that he should, as will appear upon reading the whole; "thus saith the Lord, write ye this man childless, a man that shall not prosper in his days; for no man of his seed shall prosper, sitting on the throne of David, and ruling any more in Judah". Besides, the Hebrew word rendered "childless", comes from which signifies "to make naked" or "bare" and so denotes not only such as have no children, or are bereft of them, but such as are by any providence stripped of the blessings of life, and are left bare, destitute, and unhappy, as Jechonias and his posterity were: however, the Jews have no reason to find fault with our Evangelist, since Salathiel is expressly called Jechonias's son, 1-Chronicles 3:17 either he was his proper natural son, or, to use their way of speaking, "the son of the kingdom" (e), that is, his heir and successor in the kingdom, as some have thought; since it looks as if he was the son of Neri, Luke 3:27 though the chronicle of Jedidaeus of Alexandria (f), or Philo the Jew, says, that Jechonias was called Neri, because Ner, or the lamp of David, shined in him, which had been almost extinguished.
And Salathiel begat Zorobabel. This account perfectly agrees with many passages in the Old Testament, where Zorobabel is called the son of Shealtiel or Salathiel, Ezra 3:2 Haggai 1:1 which is sufficient to justify the Evangelist in this assertion. There is indeed a difficulty which as much presses the Jews as the Christians, and that is, that Zorobabel is reckoned as the son of Pedaiah, 1-Chronicles 3:19 for the solution of which a noted Jewish commentator (g) observes, that
"in Haggai, Zachariah and Ezra, Zorobabel is called the son of Shealtiel, because he was his son's son; for Pedaiah was the son of Shealtiel, and Zorobabel the son of Pedaiah; and do not you observe (adds he) that in many places children's children are mentioned as children?''
No doubt there are many instances of this; but to me it seems that Pedaiah was not the son of Shealtiel, but his brother, 1-Chronicles 3:17. And I greatly suspect that Shealtiel had no children of his own, since none are mentioned; and that he adopted his brother Pedaiah's son Zorobabel, and made him his heir and successor in the government of Judah. However, it is certain, as a genealogical writer (h) among the Jews observes, that he was of the son's sons of Jechonias, king of Judah, from whom our Evangelist makes him to descend.
(c) Hist. Dynast. p. 45. Vid. Hieron. Comment. in Daniel. i. fol. 264. B. (d) Kimchi in 1 Chron. iii. 17. & in Jeremiah. xxii. 30. (e) Ib. in 1 Chron. iii. 15. (f) Apud Vorst. Observ. in Ganz. Chronolog. p. 310. (g) Kimchi in 1 Chron. iii. 19. & in Hagg. i. 1. (h) Juchasin, fol. 13. i.

And after they were brought to Babylon--after the migration of Babylon.
Jechonias begat Salathiel--So 1-Chronicles 3:17. Nor does this contradict Jeremiah 22:30, "Thus saith the Lord, Write ye this man (Coniah, or Jeconiah) childless"; for what follows explains in what sense this was meant--"for no man of his seed shall prosper, sitting upon the throne of David." He was to have seed, but no reigning child.
and Salathiel--or Shealtiel.
begat Zorobabel--So Ezra 3:2; Nehemiah 12:1; Haggai 1:1. But it would appear from 1-Chronicles 3:19 that Zerubbabel was Salathiel's grandson, being the son of Pedaiah, whose name, for some reason unknown, is omitted.

Zerubbabel. The descendant of the ancient kings who led the Jews back from the Captivity (Ezra 3:2).