Compare the margin references and notes.

Adam, Sheth, Enosh - That is, Adam was the father of Sheth or Seth. Seth was the father of Enosh, Enosh the father of Kenan, and so on. No notice is taken of Cain and Abel, or of any of the other sons of Adam. One line of patriarchs, from Adam to Noah, is what the historian intended to give; and to have mentioned the posterity of Cain or Abel would have been useless, as Noah was not the immediate descendant of either. Besides, all their posterity had perished in the deluge, none remaining of the Adamic family but Noah and his children; and from these all the nations of the earth sprang.
How learned must those men be who can take for a text "The first verse of the first chapter of the first book of Chronicles." and find a mystery in each name; which, in the aggregate, amounts to a full view of the original perfection, subsequent fall, consequent misery, and final restoration, of Man! O ye profound illustrators of the names of men and cities! why do ye not give us the key of your wisdom, write comments, and enlighten the world?

Adam, (a) Sheth, Enosh,
The Argument - The laws comprehend both these books in one, which the Grecians because of the length, divide into two: and they are called Chronicles, because they note briefly the history from Adam to the return from their captivity in Babylon. But these are not the books of Chronicles which are mentioned in the books of the kings of Judah and Israel, which set forth the story of both kingdoms, and later perished in the captivity, but an abridgement of the same, and were gathered by Ezra, as the Jews write after their return from Babylon. This first book contains a brief rehearsal of the children of Adam to Abraham, Isaac, Jacob and the twelve patriarchs, chiefly of Judah, and the reign of David, because Christ came from him according to the flesh. Therefore it sets forth more amply his acts both concerning civil government, and also the administration and care of things concerning religion, for the good success of which he rejoices and gives thanks to the Lord.
(a) Meaning, that Seth was Adam's son, and Enoch was Seth's son.

Adam, Seth,.... These first four verses exactly agree with the account of the antediluvian patriarchs in Genesis 5:1, the first letter in Adam is larger than usual, as a memorial, as Buxtorf (m) observes, of the first and only man, from whence mankind had their beginning, and whose history the author had undertaken to write.

This chapter, and many that follow, repeat the genealogies, or lists of fathers and children in the Bible history, and put them together, with many added. When compared with other places, there are some differences found; yet we must not therefore stumble at the word, but bless God that the things necessary to salvation are plain enough. The original of the Jewish nation is here traced from the first man that God created, and is thereby distinguished from the obscure, fabulous, and absurd origins assigned to other nations. But the nations now are all so mingled with one another, that no one nation, nor the greatest part of any, is descended entirely from any of one nation, nor the greatest part of any, is descended entirely from any of these fountains. Only this we are sure of, that God has created of one blood all nations of men; they are all descended from one Adam, one Noah. Have we not all one father? Has not one God created us? Malachi 2:10.

ADAM'S LINE TO NOAH. (1Ch. 1:1-23)
Adam, &c.--"Begat" must be understood. Only that one member of the family is mentioned, who came in the direct order of succession.

The patriarchs from Adam to Noah and his sons. - The names of the ten patriarchs of the primeval world, from the Creation to the Flood, and the three sons of Noah, are given according to Genesis 5, and grouped together without any link of connection whatever: it is assumed as known from Genesis, that the first ten names denote generations succeeding one another, and that the last three, on the contrary, are the names of brethren.

Sheth - Adam begat Sheth: and so in the following particulars. For brevity sake he only mentions their names; but the rest is easily understood out of the former books. This appears as the peculiar glory of the Jewish nation, that they alone were able to trace their pedigree from the first man that God created, which no other nation pretended to, but abused themselves and their posterity with fabulous accounts of their originals: the people of Thessaly fancying that they sprang from stones, the Athenians, that they grew out of the earth.