How does Narcissus die in The Alchemist?
Narcissus died because he was looking at his reflection too long and fell into the lake.
The alchemist finds a book that describes the legend of Narcissus.
The alchemist knew the legend of Narcissus, a youth who knelt daily beside a lake to contemplate his own beauty. He was so fascinated by himself that, one morning, he fell into the lake and drowned. (prologue)
The alchemist thinks the story’s ending is interesting because it describes a goddess who asks the lake why it weeps, and learns that the lake weeps because it can no longer look into Narcissus’s eyes and see its own beauty reflected.
The alchemist clearly sees the irony in the story. The lake where Narcissus drowned turned out to be just as vain as he was. This also reflects one of the main themes of the story, that if we get too focused on one thing we lose track of what's really important.
This is what happened to Santiago when he went looking for the treasure in his dream, only to realize that when looking for his Personal Legend and treasure, when really all he needed in life was what he had all along.