What significance do you see in Montag's choice of a passage from the Bible as the literature he will memorize?
A major theme in this novel is alienation. Mankind has alienated itself by destroying books, books which contain the record of the history of mankind. When men are alienated, they are lonely and depressed and they wind up finding no reason to live, so they commit suicide, like Mildred. It is significant that Montag decides to memorize something from the Bible, because the Bible is the word of God, given to man so that man will know how to live in the world. The particular book that Montag tries to memorize is Ecclesiastes. The reason that this is significant is that this book is written by a wise philosopher, some believe King Solomon himself. In this book, the author grapples with the meaning of life, man's place in the world, etc. The author experiences everything life has to offer (fame, fortune, family) and finds them all "meaningless, chasing after the wind." The conclusion of the book of Ecclesiastes is found in Chapter 12, verse 13:
The conclusion, when all has been heard, is: fear God and keep His commandments, because this applies to every person.
In Fahrenheit 451, man has strayed far, far away from his purpose, and thus he is totally alienated. But, there is hope - God's word will not return void and can be stored up in a man's heart through memorization, which Montag decides to do to redeem himself.
Ray Bradbury is one of my favorite authors of all time. While he often cringes when people point out his religious themes and motifs, one cannot deny their existence in almost all of his works. He was raised a Baptist so these themes are a part of the fiber of his writing, despite his denials.
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This question has been asked and answered already. Along with the answer here, see those at the link in the Sources section.