In Paulo Coelho’s The Alchemist, what are elements of alchemy that the Englishman refers to as the Elixir of Life and the Philosopher's Stone?
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In Paulo Coelho’s The Alchemist, the Englishman that travels with Santiago to the oasis in the desert is looking for the alchemist, as is Santiago. Santiago is looking for looking for the alchemist to help him find his treasure, but follows omens and learns to speak the Language of the World in this way (a way to connect with nature, the universe and other people) while searching to fulfill his Personal Legend (what he is meant to do and be in the world).
On the other hand, the Englishman is searching for the alchemist, as so many people have in the past, with the unsubstantiated belief that base, worthless metals (like lead) can "scientifically" be changed into gold, for instance. The Englishman is well-read in the subject, and carries his books across the desert. He has never had any success. He believes that Santiago is foolish for not paying attention to what the Englishman regards as "truths" in his books, but to Santiago, they have no value.
As they travel, the Englishman tries to share his knowledge with the boy who is already listening to the Language of the World and watching the omens along the way. He asks the Englishman why he cannot find his truths the same way Santiago does. The Englishman insists that they must follow the steps of alchemy exactly, as the "masters" followed those steps.
The Englishman teaches Santiago the parts of alchemy:
...the liquid part of the Master Work was called the Elixir of Life, and...it cured all illnesses; it also kept the alchemist from growing old. And the solid part was called the Philosopher's Stone.
The Englishman insists that a lifetime committed to alchemy led alchemists to give up "the vanities of the world"...and a way of finding "a purification of themselves." The Philosopher's Stone was important most especially because a very small amount of that stone could produce the transformation of enormous amounts "of metal into gold."
Of course, when the Englishman meets the alchemist, the older man tells him simply to continue to try. Without question, the Englishman does so. However, it is Santiago who finds direction from the alchemist: he does not just seek the treasure of his recurring dream, but watches for omens and pursues his Personal Legend, which the Englishman does not. In this way, Santiago will be successful, where the Englishman will fail.