"To one as young as you, I'm sure it seems incredible, but to Nicolas and Perenelle, it really is like going to bed after a very, very long day."
-- Albus Dumbledore, comforting Harry about the Flamels' approaching deaths (PS17)
The Flamels live for over six hundred years thanks to Nicolas having created the Philosopher’s Stone. After Lord Voldemort tries to get the Philosopher’s Stone, it is decided that the Stone should be destroyed. The Flamels use the last of the Elixir of Life to stay alive long enough to see that their affairs are in order, then pass away, fully accepting the end of their lives. (PS17).
We don't know when the Flamels actually passed away. We only know that with the loss of the Philospher's Stone they no longer were immortal and that sometime after June of 1992, they did/will die.
Coming to terms with and learning to accept death is a major theme in the series, and the Flamels are an interesting case. -BB
The Flamels calm, even willing, acceptance of death is very similar to Ignotus Peverell "greet[ing] Death as an old friend" (TBB). -BB
The three Peverell brothers, of course, were also possessors of objects that tried to cheat death (TBB). -BB
Dumbledore tells Harry, "You know, the Stone was really not such a wonderful thing. As much money and life as you could want! The two things most human beings would choose above all - the trouble is, humans do have a knack of choosing precisely those things that are worst for them." (PS17). Is Dumbledore just speaking about people in general here? Nicolas Flamel was a friend of Dumbledore's, and the two worked on projects together. Were Nicolas and Perenelle so unhappy? And what made them decide they could gladly and peacefully die now when they had kept themselves alive for hundreds of years? -BB
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