What is the significance of The Book of Ecclesiastes in "Fahrenheit 451"?
Most obviously, it is the book that Montag has partially memorized, and that he will, in their future society, teach to other people. Part of Granger's plan, and those that return to the city to rebuild, is to use books that they have all memorized, so that they can share the wisdom from them, and pass that wisdom on to future generations. Montag says that he has a little bit of Ecclesiastes in his head, so he will be in charge of guarding it safely so that it isn't lost forever.
More symbolically, in the actual book of Ecclesiastes in the bible, a lot of its main theme focuses on how worldy pleasures, vanity, material goods, and worldly pursuits are worthless, meaningless, and empty. Pursuing the world, filling your life with pursuits of money and pleasure will leave you without anything of real import in the end. That theme represents the society that Montag lived in, and how it is empty and meaningless, despite its full and busy schedule as it pursues worldy pleasures. That is a significant book in the bible to memorize, because it contains important lessons on how to rebuild a society that is based on real, meaningful values instead of emptiness.
As was stated in the previous response, Montag attempts to remember the book of Ecclesiastes in hopes of one day sharing it with others in a rebuilt society. Ecclesiastes was written by King Solomon who according to the Bible was the wisest man who ever lived. The book chronicles Solomon's personal pilgrimage and warns future generations not to squander God's blessings for personal glory. Throughout the book, Solomon discusses the futility of earthly pleasures and argues that an unexamined life apart from God is meaningless. Solomon's experiences and message correlates with the themes throughout Bradbury's novel. In the dystopian society, citizens seek immediate pleasure and satisfaction. They are self-centered, superficial, and live meaningless lives. Montag's ability to remember Ecclesiastes is significant so that future generations will not seek happiness in earthly pleasures. The verse that Montag recalls at the end of the novel is Ecclesiastes 3:1, "There is a time for everything, and a season for every activity under the heavens" (NIV). This verse correlates with Montag's approaching journey to rebuild a literate, intelligent society.