What is the significance of the subtitles in Fahrenheit 451?For example The Hearth and the Salamander
Each of the titles reflects Montag's development as a character. They symbolize the various stages of his rejection of society's values, and the ways in which he goes about declaring his independence, if you will.
Hearth is another word for fireplace, and the salamander is the official symbol of the firemen, as well as what they call their fire trucks. Both of these symbols have to do with fire, which dominates Montag's life at this point. As a fireman, Montag lives with fire and loves it, taking pride in his work (as the opening line points out). As far as how these images connect to this idea: one would light a fire in a hearth, and the salamander of myth lives in fire and is unaffected by flames. Thus, this section is an introduction to Montag's home and his job.
The title of the next section, “The Sieve and the Sand”, describes Montag's attempt to read and memorize the entire Bible. He connects it to a childhood memory of trying to fill a sieve with sand on the beach, which ends with him crying at the futility of the task. The sand becomes the words of the books, and the truths within them, while the sieve is the human mind. Truth is elusive and, the metaphor suggests, impossible to grasp in any permanent way.
The final section, entitled "Burning Bright", is the section where Montag fully breaks away from his former life, and chooses to live instead on the margins. He is "burning" in that his revolt is in full force, and he is stripping away the remnants of his old identity. He burns his house, he burns Captain Beatty and his co-workers, and he runs. It may also be a reference to the nuclear bomb which destroys the city at the end of the novel.